{"id":818,"date":"2018-12-11T13:22:26","date_gmt":"2018-12-11T18:22:26","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/use-the-language-of-algebra-2\/"},"modified":"2018-12-11T13:22:26","modified_gmt":"2018-12-11T18:22:26","slug":"use-the-language-of-algebra-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/use-the-language-of-algebra-2\/","title":{"raw":"Use the Language of Algebra","rendered":"Use the Language of Algebra"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Find factors, prime factorizations, and least common multiples<\/li><li>Use variables and algebraic symbols<\/li><li>Simplify expressions using the order of operations<\/li><li>Evaluate an expression<\/li><li>Identify and combine like terms<\/li><li>Translate an English phrase to an algebraic expression<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836546004\" class=\"be-prepared\"><p id=\"fs-id1167833056736\">This chapter is intended to be a brief review of concepts that will be needed in an Intermediate Algebra course. A more thorough introduction to the topics covered in this chapter can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833061637\"><h3 data-type=\"title\">Find Factors, Prime Factorizations, and Least Common Multiples<\/h3><p id=\"fs-id1167836334858\">The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A <span data-type=\"term\">multiple<\/span> of 2 can be written as the product of a counting number and 2.<\/p><span data-type=\"media\" id=\"fs-id1167836376102\" data-alt=\"Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.\"><\/span><p id=\"fs-id1167836349436\">Similarly, a multiple of 3 would be the product of a counting number and 3.<\/p><span data-type=\"media\" id=\"fs-id1167836282492\" data-alt=\"Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.\"><\/span><p id=\"fs-id1167836360898\">We could find the multiples of any number by continuing this process.<\/p><table id=\"fs-id1167836545969\" class=\"unnumbered\" summary=\"This table has 13 columns, 8 rows and a header row. The header row labels each column: counting number, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first column labels each row: multiples of 2, multiples of 3, multiples of 4, multiples of 5, multiples of 6, multiples of 7, multiples of 8, multiples of 9. The column labeled 1 has the following values: 2, 3, 4, 5, 6, 7, 8, 9. The column labeled 2 has the following values: 4, 6, 8, 10, 12, 14, 16, 18. The column labeled 3 has the following values: 6, 9, 12, 15, 18, 21, 24, 27. The column labeled 4 has the following values: 8, 12, 16, 20, 24, 28, 32, 36. The column labeled 5 has the following values: 10, 15, 20, 25, 30, 35, 40, 45. The column labeled 6 has the following values: 12, 18, 24, 30, 36, 42, 48, 54. The column labeled 7 has the following values: 14, 21, 28, 35, 42, 49, 56, 63. The column labeled 8 has the following values: 16, 24, 32, 40, 48, 56, 64, 72. The column labeled 9 has the following values: 18, 27, 36, 45, 54, 63, 72, 81. The column labeled 10 has the following values: 20, 30, 40, 50, 60, 70, 80, 90. The column labeled 11 has the following values: 22, 33, 44, 55, 66, 77, 88, 99. The column labeled 12 has the following values: 24, 36, 48, 60, 72, 84, 96, 108.\"><thead><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"left\">Counting Number<\/th><th data-valign=\"middle\" data-align=\"left\">1<\/th><th data-valign=\"middle\" data-align=\"left\">2<\/th><th data-valign=\"middle\" data-align=\"left\">3<\/th><th data-valign=\"middle\" data-align=\"left\">4<\/th><th data-valign=\"middle\" data-align=\"left\">5<\/th><th data-valign=\"middle\" data-align=\"left\">6<\/th><th data-valign=\"middle\" data-align=\"left\">7<\/th><th data-valign=\"middle\" data-align=\"left\">8<\/th><th data-valign=\"middle\" data-align=\"left\">9<\/th><th data-valign=\"middle\" data-align=\"left\">10<\/th><th data-valign=\"middle\" data-align=\"left\">11<\/th><th data-valign=\"middle\" data-align=\"left\">12<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 2<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">2<\/td><td data-valign=\"middle\" data-align=\"left\">4<\/td><td data-valign=\"middle\" data-align=\"left\">6<\/td><td data-valign=\"middle\" data-align=\"left\">8<\/td><td data-valign=\"middle\" data-align=\"left\">10<\/td><td data-valign=\"middle\" data-align=\"left\">12<\/td><td data-valign=\"middle\" data-align=\"left\">14<\/td><td data-valign=\"middle\" data-align=\"left\">16<\/td><td data-valign=\"middle\" data-align=\"left\">18<\/td><td data-valign=\"middle\" data-align=\"left\">20<\/td><td data-valign=\"middle\" data-align=\"left\">22<\/td><td data-valign=\"middle\" data-align=\"left\">24<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 3<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">3<\/td><td data-valign=\"middle\" data-align=\"left\">6<\/td><td data-valign=\"middle\" data-align=\"left\">9<\/td><td data-valign=\"middle\" data-align=\"left\">12<\/td><td data-valign=\"middle\" data-align=\"left\">15<\/td><td data-valign=\"middle\" data-align=\"left\">18<\/td><td data-valign=\"middle\" data-align=\"left\">21<\/td><td data-valign=\"middle\" data-align=\"left\">24<\/td><td data-valign=\"middle\" data-align=\"left\">27<\/td><td data-valign=\"middle\" data-align=\"left\">30<\/td><td data-valign=\"middle\" data-align=\"left\">33<\/td><td data-valign=\"middle\" data-align=\"left\">36<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 4<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">4<\/td><td data-valign=\"middle\" data-align=\"left\">8<\/td><td data-valign=\"middle\" data-align=\"left\">12<\/td><td data-valign=\"middle\" data-align=\"left\">16<\/td><td data-valign=\"middle\" data-align=\"left\">20<\/td><td data-valign=\"middle\" data-align=\"left\">24<\/td><td data-valign=\"middle\" data-align=\"left\">28<\/td><td data-valign=\"middle\" data-align=\"left\">32<\/td><td data-valign=\"middle\" data-align=\"left\">36<\/td><td data-valign=\"middle\" data-align=\"left\">40<\/td><td data-valign=\"middle\" data-align=\"left\">44<\/td><td data-valign=\"middle\" data-align=\"left\">48<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 5<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">5<\/td><td data-valign=\"middle\" data-align=\"left\">10<\/td><td data-valign=\"middle\" data-align=\"left\">15<\/td><td data-valign=\"middle\" data-align=\"left\">20<\/td><td data-valign=\"middle\" data-align=\"left\">25<\/td><td data-valign=\"middle\" data-align=\"left\">30<\/td><td data-valign=\"middle\" data-align=\"left\">35<\/td><td data-valign=\"middle\" data-align=\"left\">40<\/td><td data-valign=\"middle\" data-align=\"left\">45<\/td><td data-valign=\"middle\" data-align=\"left\">50<\/td><td data-valign=\"middle\" data-align=\"left\">55<\/td><td data-valign=\"middle\" data-align=\"left\">60<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 6<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">6<\/td><td data-valign=\"middle\" data-align=\"left\">12<\/td><td data-valign=\"middle\" data-align=\"left\">18<\/td><td data-valign=\"middle\" data-align=\"left\">24<\/td><td data-valign=\"middle\" data-align=\"left\">30<\/td><td data-valign=\"middle\" data-align=\"left\">36<\/td><td data-valign=\"middle\" data-align=\"left\">42<\/td><td data-valign=\"middle\" data-align=\"left\">48<\/td><td data-valign=\"middle\" data-align=\"left\">54<\/td><td data-valign=\"middle\" data-align=\"left\">60<\/td><td data-valign=\"middle\" data-align=\"left\">66<\/td><td data-valign=\"middle\" data-align=\"left\">72<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 7<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">7<\/td><td data-valign=\"middle\" data-align=\"left\">14<\/td><td data-valign=\"middle\" data-align=\"left\">21<\/td><td data-valign=\"middle\" data-align=\"left\">28<\/td><td data-valign=\"middle\" data-align=\"left\">35<\/td><td data-valign=\"middle\" data-align=\"left\">42<\/td><td data-valign=\"middle\" data-align=\"left\">49<\/td><td data-valign=\"middle\" data-align=\"left\">56<\/td><td data-valign=\"middle\" data-align=\"left\">63<\/td><td data-valign=\"middle\" data-align=\"left\">70<\/td><td data-valign=\"middle\" data-align=\"left\">77<\/td><td data-valign=\"middle\" data-align=\"left\">84<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 8<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">8<\/td><td data-valign=\"middle\" data-align=\"left\">16<\/td><td data-valign=\"middle\" data-align=\"left\">24<\/td><td data-valign=\"middle\" data-align=\"left\">32<\/td><td data-valign=\"middle\" data-align=\"left\">40<\/td><td data-valign=\"middle\" data-align=\"left\">48<\/td><td data-valign=\"middle\" data-align=\"left\">56<\/td><td data-valign=\"middle\" data-align=\"left\">64<\/td><td data-valign=\"middle\" data-align=\"left\">72<\/td><td data-valign=\"middle\" data-align=\"left\">80<\/td><td data-valign=\"middle\" data-align=\"left\">88<\/td><td data-valign=\"middle\" data-align=\"left\">96<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 9<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">9<\/td><td data-valign=\"middle\" data-align=\"left\">18<\/td><td data-valign=\"middle\" data-align=\"left\">27<\/td><td data-valign=\"middle\" data-align=\"left\">36<\/td><td data-valign=\"middle\" data-align=\"left\">45<\/td><td data-valign=\"middle\" data-align=\"left\">54<\/td><td data-valign=\"middle\" data-align=\"left\">63<\/td><td data-valign=\"middle\" data-align=\"left\">72<\/td><td data-valign=\"middle\" data-align=\"left\">81<\/td><td data-valign=\"middle\" data-align=\"left\">90<\/td><td data-valign=\"middle\" data-align=\"left\">99<\/td><td data-valign=\"middle\" data-align=\"left\">108<\/td><\/tr><\/tbody><\/table><div data-type=\"note\" id=\"fs-id1167836548764\"><div data-type=\"title\">Multiple of a Number<\/div><p id=\"fs-id1167836507946\">A number is a <strong data-effect=\"bold\">multiple<\/strong> of \\(n\\) if it is the product of a counting number and \\(n.\\)<\/p><\/div><p id=\"fs-id1167836700519\">Another way to say that 15 is a multiple of 3 is to say that 15 is <span data-type=\"term\">divisible<\/span> by 3. That means that when we divide 3 into 15, we get a counting number. In fact, \\(15\u00f73\\) is 5, so 15 is \\(5\u00b73.\\)<\/p><div data-type=\"note\" id=\"fs-id1167836362413\"><div data-type=\"title\">Divisible by a Number<\/div><p id=\"fs-id1167836447704\">If a number \\(m\\) is a multiple of <em data-effect=\"italics\">n<\/em>, then <em data-effect=\"italics\">m<\/em> is <strong data-effect=\"bold\">divisible<\/strong> by <em data-effect=\"italics\">n<\/em>.<\/p><\/div><p id=\"fs-id1167836415616\">If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:<\/p><div data-type=\"note\" id=\"fs-id1167836525210\"><div data-type=\"title\">Divisibility Tests<\/div><p id=\"fs-id1167829693509\">A number is divisible by:<\/p><p id=\"fs-id1167836447272\">\u2003\u2003\u20032 if the last digit is 0, 2, 4, 6, or 8.<\/p><p id=\"fs-id1167836486860\">\u2003\u2003\u20033 if the sum of the digits is divisible by \\(3.\\)<\/p><p id=\"fs-id1167836534606\">\u2003\u2003\u20035 if the last digit is 5 or \\(0.\\)<\/p><p id=\"fs-id1167836510672\">\u2003\u2003\u20036 if it is divisible by both 2 and \\(3.\\)<\/p><p id=\"fs-id1167836608078\">\u2003\u2003\u200310 if it ends with \\(0.\\)<\/p><\/div><div data-type=\"example\" id=\"fs-id1167833047554\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836516080\"><div data-type=\"problem\" id=\"fs-id1167836558662\"><p id=\"fs-id1167833053866\">Is 5,625 divisible by <span class=\"token\">\u24d0<\/span> 2? <span class=\"token\">\u24d1<\/span> 3? <span class=\"token\">\u24d2<\/span> 5 or 10? <span class=\"token\">\u24d3<\/span> 6?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836579398\"><ol id=\"fs-id1167836510387\" type=\"1\" class=\"circled\"><li><span class=\"token\">\u24d0<\/span><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Is 5,625 divisible by 2?}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\text{Does it end in 0, 2, 4, 6 or 8?}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{No.}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{5,625 is not divisible by 2.}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><\/li><li><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Is 5,625 divisible by 3?}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\text{What is the sum of the digits?}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}5+6+2+5=18\\hfill \\\\ \\text{Is the sum divisible by 3?}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{Yes.}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{5,625 is divisible by 3.}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><\/li><li><span class=\"token\">\u24d2<\/span><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Is 5,625 divisible by 5 or 10?}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\text{What is the last digit? It is 5.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{5,625 is divisible by 5 but not by 10.}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><\/li><li><span class=\"token\">\u24d3<\/span><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Is 5,625 divisible by 6?}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\text{Is it divisible by both 2 and 3?}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{No, 5,625 is not divisible by 2, so 5,625 is}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{not divisible by 6.}\\hfill \\end{array}\\)<\/li><\/ol><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836691875\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829691614\"><div data-type=\"problem\" id=\"fs-id1167836356400\"><p id=\"fs-id1167829694806\">Is 4,962 divisible by <span class=\"token\">\u24d0<\/span> 2? <span class=\"token\">\u24d1<\/span> 3? <span class=\"token\">\u24d2<\/span> 5? <span class=\"token\">\u24d3<\/span> 6? <span class=\"token\">\u24d4<\/span> 10?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836342477\"><p id=\"fs-id1167829694818\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> yes<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span> no<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833019387\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836311134\"><div data-type=\"problem\" id=\"fs-id1167836551933\"><p id=\"fs-id1167836293370\">Is 3,765 divisible by <span class=\"token\">\u24d0<\/span> 2? <span class=\"token\">\u24d1<\/span> 3? <span class=\"token\">\u24d2<\/span> 5? <span class=\"token\">\u24d3<\/span> 6? <span class=\"token\">\u24d4<\/span> 10?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836341297\"><p id=\"fs-id1167836559793\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> yes <span class=\"token\">\u24d3<\/span> no<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span> no<\/div><\/div><\/div><p id=\"fs-id1167836511506\">In mathematics, there are often several ways to talk about the same ideas. So far, we\u2019ve seen that if <em data-effect=\"italics\">m<\/em> is a multiple of <em data-effect=\"italics\">n<\/em>, we can say that <em data-effect=\"italics\">m<\/em> is divisible by <em data-effect=\"italics\">n<\/em>. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.<\/p><p id=\"fs-id1167836612806\">Since \\(8\u00b79=72,\\) we say that 8 and 9 are <span data-type=\"term\">factors<\/span> of 72. When we write \\(72=8\u00b79,\\) we say we have factored 72.<\/p><span data-type=\"media\" id=\"fs-id1167833056551\" data-alt=\"8 times 9 is 72. 8 and 9 are factors. 72 is the product.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"8 times 9 is 72. 8 and 9 are factors. 72 is the product.\"><\/span><p id=\"fs-id1167836387255\">Other ways to factor 72 are \\(1\u00b772,\\phantom{\\rule{0.8em}{0ex}}2\u00b736,\\phantom{\\rule{0.8em}{0ex}}3\u00b724,\\phantom{\\rule{0.8em}{0ex}}4\u00b718,\\) and \\(6\u00b712.\\) The number 72 has many factors: \\(1,2,3,4,6,8,9,12,18,24,36,\\) and \\(72.\\)<\/p><div data-type=\"note\" id=\"fs-id1167836539545\"><div data-type=\"title\">Factors<\/div><p id=\"fs-id1167836322020\">If \\(a\u00b7b=m,\\) then <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are <strong data-effect=\"bold\">factors<\/strong> of <em data-effect=\"italics\">m<\/em>.<\/p><\/div><p id=\"fs-id1167836387232\">Some numbers, such as 72, have many factors. Other numbers have only two factors. A <span data-type=\"term\">prime number<\/span> is a counting number greater than 1 whose only factors are 1 and itself.<\/p><div data-type=\"note\" id=\"fs-id1167836410498\"><div data-type=\"title\">Prime number and Composite number<\/div><p id=\"fs-id1167836538134\">A <strong data-effect=\"bold\">prime number<\/strong> is a counting number greater than 1 whose only factors are 1 and the number itself.<\/p><p id=\"fs-id1167829695314\">A <strong data-effect=\"bold\">composite number<\/strong> is a counting number that is not prime. A composite number has factors other than 1 and the number itself.<\/p><\/div><p id=\"fs-id1167836515162\">The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the \u201cprime\u201d or \u201ccomposite\u201d label for each!<\/p><span data-type=\"media\" id=\"fs-id1167836418575\" data-alt=\"This table has three columns, 19 rows and a header row. The header row labels each column: number, factors and prime or composite. The values in each row are as follows: number 2, factors 1, 2, prime; number 3, factors 1, 3, prime; number 4, factors 1, 2, 4, composite; number 5, factors, 1, 5, prime; number 6, factors 1, 2, 3, 6, composite; number 7, factors 1, 7, prime; number 8, factors 1, 2, 4, 8, composite; number 9, factors 1, 3, 9, composite; number 10, factors 1, 2, 5, 10, composite; number 11, factors 1, 11, prime; number 12, factors 1, 2, 3, 4, 6, 12, composite; number 13, factors 1, 13, prime; number 14, factors 1, 2, 7, 14, composite; number 15, factors 1, 3, 5, 15, composite; number 16, factors 1, 2, 4, 8, 16, composite; number 17, factors 1, 17, prime; number 18, factors 1, 2, 3, 6, 9, 18, composite; number 19, factors 1, 19, prime; number 20, factors 1, 2, 4, 5, 10, 20, composite.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_004_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has three columns, 19 rows and a header row. The header row labels each column: number, factors and prime or composite. The values in each row are as follows: number 2, factors 1, 2, prime; number 3, factors 1, 3, prime; number 4, factors 1, 2, 4, composite; number 5, factors, 1, 5, prime; number 6, factors 1, 2, 3, 6, composite; number 7, factors 1, 7, prime; number 8, factors 1, 2, 4, 8, composite; number 9, factors 1, 3, 9, composite; number 10, factors 1, 2, 5, 10, composite; number 11, factors 1, 11, prime; number 12, factors 1, 2, 3, 4, 6, 12, composite; number 13, factors 1, 13, prime; number 14, factors 1, 2, 7, 14, composite; number 15, factors 1, 3, 5, 15, composite; number 16, factors 1, 2, 4, 8, 16, composite; number 17, factors 1, 17, prime; number 18, factors 1, 2, 3, 6, 9, 18, composite; number 19, factors 1, 19, prime; number 20, factors 1, 2, 4, 5, 10, 20, composite.\"><\/span><p id=\"fs-id1167829694566\">The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.<\/p><p id=\"fs-id1167833060885\">A composite number can be written as a unique product of primes. This is called the <span data-type=\"term\">prime factorization<\/span> of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.<\/p><div data-type=\"note\" id=\"fs-id1167836530500\"><div data-type=\"title\">Prime Factorization<\/div><p id=\"fs-id1167836628445\">The <strong data-effect=\"bold\">prime factorization<\/strong> of a number is the product of prime numbers that equals the number.<\/p><\/div><p id=\"fs-id1167836546976\">To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.<\/p><p id=\"fs-id1167836334529\">If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.<\/p><div data-type=\"example\" id=\"fs-id1167829937177\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Find the Prime Factorization of a Composite Number<\/div><div data-type=\"exercise\" id=\"fs-id1167829599878\"><div data-type=\"problem\" id=\"fs-id1167829599880\"><p id=\"fs-id1167829599883\">Factor 48.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833056756\"><span data-type=\"media\" id=\"fs-id1167833056758\" data-alt=\"Step 1 is to find two factors whose product is 48 and use these numbers to create two branches. The two branches originating from 48 are formed by the factors 2 and 24.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find two factors whose product is 48 and use these numbers to create two branches. The two branches originating from 48 are formed by the factors 2 and 24.\"><\/span><span data-type=\"media\" id=\"fs-id1167836542295\" data-alt=\"Step 2 is to circle the prime factor. This completes that branch. In this case, 2 is circled as it is prime.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to circle the prime factor. This completes that branch. In this case, 2 is circled as it is prime.\"><\/span><span data-type=\"media\" id=\"fs-id1167836513367\" data-alt=\"Step 3 is to treat the composite factor as a product, break it into two more factors and continue the process. 24 is not prime. It is broken into 4 and 6. 4 and 6 are not prime. 4 is broken into its factors 2 and 2, both of which are circled. 6 is not prime. It is broken into factors 2 and 3, both of which are circled.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to treat the composite factor as a product, break it into two more factors and continue the process. 24 is not prime. It is broken into 4 and 6. 4 and 6 are not prime. 4 is broken into its factors 2 and 2, both of which are circled. 6 is not prime. It is broken into factors 2 and 3, both of which are circled.\"><\/span><span data-type=\"media\" id=\"fs-id1167836507776\" data-alt=\"Step 4 is to write the original composite number as the product of all the circled primes. 48 is 2 into 2 into 2 into 2 into 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to write the original composite number as the product of all the circled primes. 48 is 2 into 2 into 2 into 2 into 3.\"><\/span><p id=\"fs-id1171790680445\"><\/p><div data-type=\"newline\"><br><\/div><p id=\"fs-id1167836545840\">We say \\(2\u00b72\u00b72\u00b72\u00b73\\) is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.<\/p><p id=\"fs-id1167836516314\">If we first factored 48 in a different way, for example as \\(6\u00b78,\\) the result would still be the same. Finish the prime factorization and verify this for yourself.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836506869\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836506872\"><div data-type=\"problem\" id=\"fs-id1167836558433\"><p id=\"fs-id1167836558436\">Find the prime factorization of \\(80.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829694799\"><p id=\"fs-id1167829694801\">\\(2\u00b72\u00b72\u00b72\u00b75\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836546878\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836418325\"><div data-type=\"problem\" id=\"fs-id1167836418327\"><p id=\"fs-id1167836418329\">Find the prime factorization of \\(60.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836418464\"><p id=\"fs-id1167836418466\">\\(2\u00b72\u00b73\u00b75\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836386665\" class=\"howto\"><div data-type=\"title\">Find the prime factorization of a composite number.<\/div><ol id=\"fs-id1167836610439\" type=\"1\" class=\"stepwise\"><li>Find two factors whose product is the given number, and use these numbers to create two branches.<\/li><li>If a factor is prime, that branch is complete. Circle the prime, like a leaf on the tree.<\/li><li>If a factor is not prime, write it as the product of two factors and continue the process.<\/li><li>Write the composite number as the product of all the circled primes.<\/li><\/ol><\/div><p id=\"fs-id1167836693029\">One of the reasons we look at primes is to use these techniques to find the <span data-type=\"term\">least common multiple<\/span> of two numbers. This will be useful when we add and subtract fractions with different denominators.<\/p><div data-type=\"note\" id=\"fs-id1167836287670\"><div data-type=\"title\">Least Common Multiple<\/div><p id=\"fs-id1167833009881\">The <strong data-effect=\"bold\">least common multiple (LCM)<\/strong> of two numbers is the smallest number that is a multiple of both numbers.<\/p><\/div><p id=\"fs-id1167836558236\">To find the least common multiple of two numbers we will use the Prime Factors Method. Let\u2019s find the LCM of 12 and 18 using their prime factors.<\/p><div data-type=\"example\" id=\"fs-id1167829937221\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Find the Least Common Multiple Using the Prime Factors Method<\/div><div data-type=\"exercise\" id=\"fs-id1167836506713\"><div data-type=\"problem\" id=\"fs-id1167836506715\"><p id=\"fs-id1167829691676\">Find the least common multiple (LCM) of 12 and 18 using the prime factors method.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829691681\"><span data-type=\"media\" id=\"fs-id1167836610203\" data-alt=\"Step 1 is to write each number as a product of primes. The number 12 is written as a product of 2, 2 and 3. The number 18 is written as a product of 2, 3 and 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write each number as a product of primes. The number 12 is written as a product of 2, 2 and 3. The number 18 is written as a product of 2, 3 and 3.\"><\/span><span data-type=\"media\" id=\"fs-id1167829692860\" data-alt=\"Step 2 is to list the primes of each number such that primes are vertically matched when possible. The factors of 12 are listed as 2, 2 and 3. The factors of 18 are written below this. The first 2 at the top lines up with the first two at the bottom. The second 2 at the top does not line up with anything. The 3 at the top lines up with a 3 at the bottom. The last 3 at the bottom does not line up with anything. Hence, four columns are made.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to list the primes of each number such that primes are vertically matched when possible. The factors of 12 are listed as 2, 2 and 3. The factors of 18 are written below this. The first 2 at the top lines up with the first two at the bottom. The second 2 at the top does not line up with anything. The 3 at the top lines up with a 3 at the bottom. The last 3 at the bottom does not line up with anything. Hence, four columns are made.\"><\/span><span data-type=\"media\" id=\"fs-id1167836622827\" data-alt=\"Step 3 is to bring down the number from each column. When a column has the same number at the top and the bottom, that number is brought down. When a column has only one number that number is brought down. The numbers brought down are 2, 2, 3 and 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to bring down the number from each column. When a column has the same number at the top and the bottom, that number is brought down. When a column has only one number that number is brought down. The numbers brought down are 2, 2, 3 and 3.\"><\/span><span data-type=\"media\" id=\"fs-id1167836608172\" data-alt=\"Step 4 is to multiply the factors. The numbers brought down are multiplied with each other to get the LCM. The LCM is 2 into 2 into 3 into 3 equal to 36.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to multiply the factors. The numbers brought down are multiplied with each other to get the LCM. The LCM is 2 into 2 into 3 into 3 equal to 36.\"><\/span><\/div><\/div><\/div><p id=\"fs-id1167836387125\">Notice that the prime factors of 12 \\(\\left(2\u00b72\u00b73\\right)\\) and the prime factors of 18 \\(\\left(2\u00b73\u00b73\\right)\\) are included in the LCM \\(\\left(2\u00b72\u00b73\u00b73\\right).\\) So 36 is the least common multiple of 12 and 18.<\/p><p id=\"fs-id1167829691476\">By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the <em data-effect=\"italics\">least<\/em> common multiple.<\/p><div data-type=\"note\" id=\"fs-id1167836693644\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836693647\"><div data-type=\"problem\" id=\"fs-id1167836693649\"><p id=\"fs-id1167836556107\">Find the LCM of 9 and 12 using the Prime Factors Method.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836556111\"><p id=\"fs-id1167836447731\">\\(36\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836556037\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836706707\"><div data-type=\"problem\" id=\"fs-id1167836706710\"><p id=\"fs-id1167836706712\">Find the LCM of 18 and 24 using the Prime Factors Method.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833279842\"><p id=\"fs-id1167833279844\">\\(72\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829597058\" class=\"howto\"><div data-type=\"title\">Find the least common multiple using the Prime Factors Method.<\/div><ol id=\"fs-id1167836317446\" type=\"1\" class=\"stepwise\"><li>Write each number as a product of primes.<\/li><li>List the primes of each number. Match primes vertically when possible.<\/li><li>Bring down the columns.<\/li><li>Multiply the factors.<\/li><\/ol><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836514368\"><h3 data-type=\"title\">Use Variables and Algebraic Symbols<\/h3><p id=\"fs-id1167836550860\">In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a <span data-type=\"term\">variable<\/span> and letters commonly used for variables are \\(x,y,a,b,c.\\)<\/p><div data-type=\"note\" id=\"fs-id1167829586674\"><div data-type=\"title\">Variable<\/div><p id=\"fs-id1167836558388\">A <strong data-effect=\"bold\">variable<\/strong> is a letter that represents a number whose value may change.<\/p><\/div><p id=\"fs-id1167829694184\">A number whose value always remains the same is called a <span data-type=\"term\">constant<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836597954\"><div data-type=\"title\">Constant<\/div><p id=\"fs-id1167833023164\">A <strong data-effect=\"bold\">constant<\/strong> is a number whose value always stays the same.<\/p><\/div><p id=\"fs-id1167833383070\">To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We\u2019ll list the symbols used to indicate these operations below.<\/p><div data-type=\"note\" id=\"fs-id1167833383073\"><div data-type=\"title\">Operation Symbols<\/div><table id=\"fs-id1167836551950\" class=\"unnumbered\" summary=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: operation, notation, say and the result is. Row 1 has the following entries: addition, a plus b, a plus b and the sum of a and b. Row 2 has the following entries: subtraction, a minus b, a minus b and the difference of a and b. Row 3 has the following entries: multiplication, notations a dot b, ab, open parentheses a close parentheses open parentheses b close parentheses, open parentheses a close parentheses b, a open parentheses b close parentheses. say a times b, the product of a and b. Row 4 has the following entries: division, a divided by b, a slash b, a upon b and b right parentheses a overbar, say a divided by b, the quotient of a and b; a is called the dividend, and b is called the divisor.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Operation<\/th><th data-valign=\"top\" data-align=\"left\">Notation<\/th><th data-valign=\"top\" data-align=\"left\">Say:<\/th><th data-valign=\"top\" data-align=\"left\">The result is\u2026<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Addition<\/td><td data-valign=\"top\" data-align=\"left\">\\(a+b\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(a\\) plus \\(b\\)<\/td><td data-valign=\"top\" data-align=\"left\">the sum of \\(a\\) and \\(b\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtraction<\/td><td data-valign=\"top\" data-align=\"left\">\\(a-b\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(a\\) minus \\(b\\)<\/td><td data-valign=\"top\" data-align=\"left\">the difference of \\(a\\) and \\(b\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiplication<\/td><td data-valign=\"top\" data-align=\"left\">\\(a\u00b7b,ab,\\left(a\\right)\\left(b\\right),\\)\\(\\left(a\\right)b,a\\left(b\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(a\\) times \\(b\\)<\/td><td data-valign=\"top\" data-align=\"left\">the product of \\(a\\) and \\(b\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Division<\/td><td data-valign=\"top\" data-align=\"left\">\\(a\u00f7b,a\\text{\/}b,\\frac{a}{b},\\phantom{\\rule{0.2em}{0ex}}b\\overline{)a}\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(a\\) divided by \\(b\\)<\/td><td data-valign=\"top\" data-align=\"left\">the quotient of \\(a\\) and \\(b;\\)<div data-type=\"newline\"><br><\/div>\\(a\\) is called the dividend, and \\(b\\) is called the divisor<\/td><\/tr><\/tbody><\/table><\/div><p id=\"fs-id1167836558375\">When two quantities have the same value, we say they are equal and connect them with an <span data-type=\"term\" class=\"no-emphasis\">equal<\/span> sign.<\/p><div data-type=\"note\" id=\"fs-id1167836614086\"><div data-type=\"title\">Equality Symbol<\/div><p id=\"fs-id1167836614092\">\\(a=b\\) is read \u201c<em data-effect=\"italics\">a<\/em> is equal to <em data-effect=\"italics\">b<\/em>.\u201d<\/p><p id=\"fs-id1167836531940\">The symbol \u201c=\u201d is called the equal sign.<\/p><\/div><p id=\"fs-id1167836531945\">On the <span data-type=\"term\" class=\"no-emphasis\">number line<\/span>, the numbers get larger as they go from left to right. The number line can be used to explain the symbols \u201c&lt;\u201d and \u201c&gt;\u201d.<\/p><div data-type=\"note\" id=\"fs-id1167833056435\"><div data-type=\"title\">Inequality<\/div><span data-type=\"media\" id=\"fs-id1171791260555\" data-alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_019_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\"><\/span><\/div><p id=\"fs-id1167833397294\">The expressions \\(a&lt;b\\) or \\(a&gt;b\\) can be read from left to right or right to left, though in English we usually read from left to right. In general,<\/p><div data-type=\"equation\" id=\"fs-id1167836624800\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}a&lt;b\\phantom{\\rule{1em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}b&gt;a.\\phantom{\\rule{0.2em}{0ex}}\\text{For example,}\\phantom{\\rule{0.2em}{0ex}}7&lt;11\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}11&gt;7.\\hfill \\\\ a&gt;b\\phantom{\\rule{1em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}b&lt;a.\\phantom{\\rule{0.2em}{0ex}}\\text{For example,}\\phantom{\\rule{0.2em}{0ex}}17&gt;4\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}4&lt;17.\\hfill \\end{array}\\)<\/div><div data-type=\"note\" id=\"fs-id1167836618823\"><div data-type=\"title\">Inequality Symbols<\/div><table id=\"fs-id1167836510455\" class=\"unnumbered\" summary=\"The table describes inequality symbols in words. The symbols described are a is not equal to b, a is less than b, a is less than or equal to b, a is greater then b, a is greater than or equal to b.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Inequality Symbols<\/th><th data-valign=\"top\" data-align=\"left\">Words<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a\\ne b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">not equal to b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a&lt;b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a\\le b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than or equal to b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a&gt;b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a\\ge b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than or equal to b.<\/em><\/td><\/tr><\/tbody><\/table><\/div><p id=\"fs-id1167836447550\">Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an <span data-type=\"term\">expression<\/span>, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.<\/p><div data-type=\"note\" id=\"fs-id1167829597232\"><div data-type=\"title\">Grouping Symbols<\/div><div data-type=\"equation\" id=\"fs-id1171790895306\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\text{Parentheses}\\hfill &amp; &amp; &amp; &amp; &amp; \\left(\\phantom{\\rule{0.2em}{0ex}}\\right)\\hfill \\\\ \\text{Brackets}\\hfill &amp; &amp; &amp; &amp; &amp; \\left[\\phantom{\\rule{0.2em}{0ex}}\\right]\\hfill \\\\ \\text{Braces}\\hfill &amp; &amp; &amp; &amp; &amp; \\left\\{\\phantom{\\rule{0.2em}{0ex}}\\right\\}\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1167836521188\">Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.<\/p><div data-type=\"equation\" id=\"fs-id1167836521192\" class=\"unnumbered\" data-label=\"\">\\(8\\left(14-8\\right)\\phantom{\\rule{5em}{0ex}}21-3\\left[2+4\\left(9-8\\right)\\right]\\phantom{\\rule{5em}{0ex}}24\u00f7\\left\\{13-2\\left[1\\left(6-5\\right)+4\\right]\\right\\}\\)<\/div><p id=\"fs-id1167833412639\">What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have <em data-effect=\"italics\">expressions<\/em> and <em data-effect=\"italics\">equations<\/em>.<\/p><div data-type=\"note\" id=\"fs-id1167833412655\"><div data-type=\"title\">Expression<\/div><p id=\"fs-id1167836530664\">An <strong data-effect=\"bold\">expression<\/strong> is a number, a variable, or a combination of numbers and variables using operation symbols.<\/p><\/div><div data-type=\"equation\" id=\"fs-id1167826987885\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccccccc}\\mathbf{\\text{Expression}}\\hfill &amp; &amp; &amp; &amp; &amp; \\mathbf{\\text{Words}}\\hfill &amp; &amp; &amp; &amp; &amp; \\mathbf{\\text{English Phrase}}\\hfill \\\\ 3+5\\hfill &amp; &amp; &amp; &amp; &amp; \\text{3 plus 5}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{the sum of three and five}\\hfill \\\\ n-1\\hfill &amp; &amp; &amp; &amp; &amp; n\\phantom{\\rule{0.2em}{0ex}}\\text{minus one}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{the difference of}\\phantom{\\rule{0.2em}{0ex}}n\\phantom{\\rule{0.2em}{0ex}}\\text{and one}\\hfill \\\\ 6\u00b77\\hfill &amp; &amp; &amp; &amp; &amp; \\text{6 times 7}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{the product of six and seven}\\hfill \\\\ \\frac{x}{y}\\hfill &amp; &amp; &amp; &amp; &amp; x\\phantom{\\rule{0.2em}{0ex}}\\text{divided by}\\phantom{\\rule{0.2em}{0ex}}y\\hfill &amp; &amp; &amp; &amp; &amp; \\text{the quotient of}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}y\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836524937\">Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.<\/p><p id=\"fs-id1167836524941\">An <span data-type=\"term\">equation<\/span> is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.<\/p><div data-type=\"note\" id=\"fs-id1167836510026\"><div data-type=\"title\">Equation<\/div><p id=\"fs-id1167836510031\">An <strong data-effect=\"bold\">equation<\/strong> is two expressions connected by an equal sign.<\/p><\/div><div data-type=\"equation\" id=\"fs-id1167835303120\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\mathbf{\\text{Equation}}\\hfill &amp; &amp; &amp; &amp; &amp; \\mathbf{\\text{English Sentence}}\\hfill \\\\ 3+5=8\\hfill &amp; &amp; &amp; &amp; &amp; \\text{The sum of three and five is equal to eight.}\\hfill \\\\ n-1=14\\hfill &amp; &amp; &amp; &amp; &amp; n\\phantom{\\rule{0.2em}{0ex}}\\text{minus one equals fourteen.}\\hfill \\\\ 6\u00b77=42\\hfill &amp; &amp; &amp; &amp; &amp; \\text{The product of six and seven is equal to forty-two.}\\hfill \\\\ x=53\\hfill &amp; &amp; &amp; &amp; &amp; x\\phantom{\\rule{0.2em}{0ex}}\\text{is equal to fifty-three.}\\hfill \\\\ y+9=2y-3\\hfill &amp; &amp; &amp; &amp; &amp; y\\phantom{\\rule{0.2em}{0ex}}\\text{plus nine is equal to two}\\phantom{\\rule{0.2em}{0ex}}y\\phantom{\\rule{0.2em}{0ex}}\\text{minus three.}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836493942\">Suppose we need to multiply 2 nine times. We could write this as \\(2\u00b72\u00b72\u00b72\u00b72\u00b72\u00b72\u00b72\u00b72.\\) This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write \\(2\u00b72\u00b72\\) as \\({2}^{3}\\) and \\(2\u00b72\u00b72\u00b72\u00b72\u00b72\u00b72\u00b72\u00b72\\) as \\({2}^{9}.\\) In expressions such as \\({2}^{3},\\) the 2 is called the <em data-effect=\"italics\">base<\/em> and the 3 is called the <em data-effect=\"italics\">exponent<\/em>. The <span data-type=\"term\" class=\"no-emphasis\">exponent<\/span> tells us how many times we need to multiply the <span data-type=\"term\" class=\"no-emphasis\">base<\/span>.<\/p><span data-type=\"media\" id=\"fs-id1167836321231\" data-alt=\"The expression shows the number 2, with the number 3 written to its top right. 2 is labeled base and 3 is labeled exponent. This means multiply 2 by itself, three times, as in 2 times 2 times 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_007_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression shows the number 2, with the number 3 written to its top right. 2 is labeled base and 3 is labeled exponent. This means multiply 2 by itself, three times, as in 2 times 2 times 2.\"><\/span><div data-type=\"note\" id=\"fs-id1167833059322\"><div data-type=\"title\">Exponential Notation<\/div><p id=\"fs-id1167833059328\">We say \\({2}^{3}\\) is in <em data-effect=\"italics\">exponential notation<\/em> and \\(2\u00b72\u00b72\\) is in <em data-effect=\"italics\">expanded notation<\/em>.<\/p><p id=\"fs-id1167836286079\">\\({a}^{n}\\) means multiply <em data-effect=\"italics\">a<\/em> by itself, <em data-effect=\"italics\">n<\/em> times.<\/p><span data-type=\"media\" id=\"fs-id1167833059485\" data-alt=\"The expression shown is a to the nth power. Here a is the base and n is the exponent. This is equal to a times a times a and so on, repeated n times. This has n factors.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_008_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression shown is a to the nth power. Here a is the base and n is the exponent. This is equal to a times a times a and so on, repeated n times. This has n factors.\"><\/span><p id=\"fs-id1167836329008\">The expression \\({a}^{n}\\) is read <em data-effect=\"italics\">a<\/em> to the \\({n}^{th}\\) power.<\/p><\/div><p id=\"fs-id1167836509491\">While we read \\({a}^{n}\\) as \\(\u201ca\\) to the \\({n}^{th}\\) power\u201d, we usually read:<\/p><div data-type=\"equation\" id=\"fs-id1167829597038\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}{a}^{2}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{\u201c}a\\phantom{\\rule{0.2em}{0ex}}\\text{squared\u201d}\\hfill \\\\ {a}^{3}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{\u201c}a\\phantom{\\rule{0.2em}{0ex}}\\text{cubed\u201d}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836493073\">We\u2019ll see later why \\({a}^{2}\\) and \\({a}^{3}\\) have special names.<\/p><p id=\"fs-id1167836493093\"><a href=\"#fs-id1167833021966\" class=\"autogenerated-content\">(Figure)<\/a> shows how we read some expressions with exponents.<\/p><table id=\"fs-id1167833021966\" summary=\"This table shows four expressions and words to describe these. The expressions described are 7 to the second power or 7 squared, 5 to the third power or 5 cubed, 9 to the fourth power and 12 to the fifth.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Expression<\/th><th data-valign=\"top\" data-align=\"left\">In Words<\/th><th data-valign=\"top\" data-align=\"left\"><\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">7<sup>2<\/sup><\/td><td data-valign=\"top\" data-align=\"left\">7 to the second power or<\/td><td data-valign=\"top\" data-align=\"left\">7 squared<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">5<sup>3<\/sup><\/td><td data-valign=\"top\" data-align=\"left\">5 to the third power or<\/td><td data-valign=\"top\" data-align=\"left\">5 cubed<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">9<sup>4<\/sup><\/td><td data-valign=\"top\" data-align=\"left\">9 to the fourth power<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">12<sup>5<\/sup><\/td><td data-valign=\"top\" data-align=\"left\">12 to the fifth power<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836416285\"><h3 data-type=\"title\">Simplify Expressions Using the Order of Operations<\/h3><p id=\"fs-id1167836416290\">To <span data-type=\"term\">simplify an expression<\/span> means to do all the math possible. For example, to simplify \\(4\u00b72+1\\) we would first multiply \\(4\u00b72\\) to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:<\/p><div data-type=\"equation\" id=\"fs-id1167836507828\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill 4\u00b72+1\\hfill \\\\ \\hfill 8+1\\hfill \\\\ \\hfill 9\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836534471\">By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.<\/p><div data-type=\"note\" id=\"fs-id1167836534475\"><div data-type=\"title\">Simplify an Expression<\/div><p id=\"fs-id1167836534481\">To <strong data-effect=\"bold\">simplify an expression<\/strong>, do all operations in the expression.<\/p><\/div><p id=\"fs-id1167836534490\">We\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the <span data-type=\"term\">order of operations<\/span>. Otherwise, expressions may have different meanings, and they may result in different values.<\/p><p id=\"fs-id1167836525166\">For example, consider the expression \\(4+3\u00b77.\\) Some students simplify this getting 49, by adding \\(4+3\\) and then multiplying that result by 7. Others get 25, by multiplying \\(3\u00b77\\) first and then adding 4.<\/p><p id=\"fs-id1167836522816\">The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.<\/p><div data-type=\"note\" id=\"fs-id1167836522820\" class=\"howto\"><div data-type=\"title\">Use the order of operations.<\/div><ol id=\"fs-id1167836522828\" type=\"1\" class=\"stepwise\"><li>Parentheses and Other Grouping Symbols <ul id=\"fs-id1167829590631\" data-bullet-style=\"bullet\"><li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li><\/ul><\/li><li>Exponents <ul id=\"fs-id1167829590645\" data-bullet-style=\"bullet\"><li>Simplify all expressions with exponents.<\/li><\/ul><\/li><li>Multiplication and Division <ul id=\"fs-id1167836375957\" data-bullet-style=\"bullet\"><li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li><\/ul><\/li><li>Addition and Subtraction <ul id=\"fs-id1167836375971\" data-bullet-style=\"bullet\"><li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li><\/ul><\/li><\/ol><\/div><p id=\"fs-id1167836375984\">Students often ask, \u201cHow will I remember the order?\u201d Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase \u201cPlease Excuse My Dear Aunt Sally\u201d.<\/p><div data-type=\"equation\" id=\"fs-id1166502309552\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\mathbf{\\text{P}}\\text{arentheses}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\mathbf{\\text{P}}\\text{lease}\\hfill \\\\ \\mathbf{\\text{E}}\\text{xponents}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\mathbf{\\text{E}}\\text{xcuse}\\hfill \\\\ \\mathbf{\\text{M}}\\text{ultiplication}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{D}}\\text{ivision}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\mathbf{\\text{M}}\\text{y}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{D}}\\text{ear}\\hfill \\\\ \\mathbf{\\text{A}}\\text{ddition}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{S}}\\text{ubtraction}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\mathbf{\\text{A}}\\text{unt}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{S}}\\text{ally}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836619914\">It\u2019s good that \u201c<strong data-effect=\"bold\">M<\/strong>y <strong data-effect=\"bold\">D<\/strong>ear\u201d goes together, as this reminds us that <strong data-effect=\"bold\">m<\/strong>ultiplication and <strong data-effect=\"bold\">d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.<\/p><p id=\"fs-id1167829597218\">Similarly, \u201c<strong data-effect=\"bold\">A<\/strong>unt <strong data-effect=\"bold\">S<\/strong>ally\u201d goes together and so reminds us that <strong data-effect=\"bold\">a<\/strong>ddition and <strong data-effect=\"bold\">s<\/strong>ubtraction also have equal priority and we do them in order from left to right.<\/p><div data-type=\"example\" id=\"fs-id1167836508031\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836508033\"><div data-type=\"problem\" id=\"fs-id1167836508035\"><p id=\"fs-id1167836508037\">Simplify: \\(18\u00f76+4\\left(5-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829692917\"><table id=\"fs-id1167829692920\" class=\"unnumbered unstyled can-break\" summary=\"The expression is 18 divided by 6 plus 4 open parentheses 5 minus 2 close parentheses. Since there are parentheses, we first open them by performing the subtraction 5 minus 2. The expression now is 18 divided by 6 plus 4 times 3. There are no exponents. Next we check for multiplication and division. Divide first because we multiply and divide left to right. We now have 3 plus 4 times 3. Next we multiply. We now have 3 plus 12. There is no other multiplication or division. Finally, we check for addition or subtraction. We add to get the number 15.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829596519\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Parentheses? Yes, subtract first.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829596546\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Exponents? No.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiplication or division? Yes.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide first because we multiply and divide left to right.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836508078\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Any other multiplication or division? Yes.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829695213\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Any other multiplication of division? No.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Any addition or subtraction? Yes.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836531883\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836698551\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836698555\"><div data-type=\"problem\" id=\"fs-id1167836698557\"><p id=\"fs-id1167836698559\">Simplify: \\(30\u00f75+10\\left(3-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836627072\"><p id=\"fs-id1167836627074\">16<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836627080\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836627083\"><div data-type=\"problem\" id=\"fs-id1167836627085\"><p id=\"fs-id1167836627088\">Simplify: \\(70\u00f710+4\\left(6-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829586618\"><p id=\"fs-id1167829586620\">23<\/p><\/div><\/div><\/div><p id=\"fs-id1167829586626\">When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.<\/p><div data-type=\"example\" id=\"fs-id1167829586631\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829586633\"><div data-type=\"problem\" id=\"fs-id1167829586635\"><p id=\"fs-id1167829586637\">Simplify: \\(5+{2}^{3}+3\\left[6-3\\left(4-2\\right)\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833056784\"><table id=\"fs-id1167833056787\" class=\"unnumbered unstyled can-break\" summary=\"The expression is 5 plus 2 to the power 3 plus 3 open bracket 6 minus 3 open parentheses 4 minus 2 close parentheses close bracket. Focus on the parentheses that are inside the brackets. Subtract to get 5 plus 2 to the power 3 plus 3 open bracket 6 minus 3 open parentheses 2 close parentheses close bracket. Continue inside the brackets and multiply to get 5 plus 2 to the power 3 plus 3 open bracket 6 minus 6 close bracket. Continue inside the brackets and subtract to get 5 plus 2 to the power 3 plus 3 open bracket 0 close bracket. The expression inside the brackets requires no further simplification. Now simplify exponents to get 5 plus 8 plus 3 open bracket 0 close bracket. Check for multiplication or division. Multiply to get 5 plus 8 plus 0. Check for addition or subtraction. Finally add to get 13.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829694073\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Are there any parentheses (or other<div data-type=\"newline\"><br><\/div>grouping symbols)? Yes.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829694118\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Focus on the parentheses that are inside the<div data-type=\"newline\"><br><\/div>brackets. Subtract.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836418416\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Continue inside the brackets and multiply.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836418442\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Continue inside the brackets and subtract.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836533805\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The expression inside the brackets requires<div data-type=\"newline\"><br><\/div>no further simplification.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Are there any exponents? Yes. Simplify exponents.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836533832\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Is there any multiplication or division? Yes.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836627134\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Is there any addition of subtraction? Yes.<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836627175\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829693866\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829693883\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829693887\"><div data-type=\"problem\" id=\"fs-id1167829693889\"><p id=\"fs-id1167829693891\">Simplify: \\(9+{5}^{3}-\\left[4\\left(9+3\\right)\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836552370\"><p id=\"fs-id1167836552372\">86<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836552378\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836552382\"><div data-type=\"problem\" id=\"fs-id1167836552385\"><p id=\"fs-id1167836552387\">Simplify: \\({7}^{2}-2\\left[4\\left(5+1\\right)\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836552426\"><p id=\"fs-id1167836552428\">1<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836530230\"><h3 data-type=\"title\">Evaluate an Expression<\/h3><p id=\"fs-id1167836530236\">In the last few examples, we simplified expressions using the order of operations. Now we\u2019ll evaluate some expressions\u2014again following the order of operations. To <span data-type=\"term\">evaluate an expression<\/span> means to find the value of the expression when the variable is replaced by a given number.<\/p><div data-type=\"note\" id=\"fs-id1167836530245\"><div data-type=\"title\">Evaluate an Expression<\/div><p id=\"fs-id1167836530250\">To <strong data-effect=\"bold\">evaluate an expression<\/strong> means to find the value of the expression when the variable is replaced by a given number.<\/p><\/div><p id=\"fs-id1167836530261\">To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.<\/p><div data-type=\"example\" id=\"fs-id1167836530265\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836530267\"><div data-type=\"problem\" id=\"fs-id1167836530269\"><p id=\"fs-id1167836530271\">Evaluate when \\(x=4:\\) <span class=\"token\">\u24d0<\/span> \\({x}^{2}\\) <span class=\"token\">\u24d1<\/span> \\({3}^{x}\\) <span class=\"token\">\u24d2<\/span> \\(2{x}^{2}+3x+8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836512101\"><p id=\"fs-id1171791331671\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836512118\" class=\"unnumbered unstyled\" summary=\"The expression is x squared. Replace x with 4 to get 4 squared. Use definition of exponent to get 4 times 4. Simplify to get 16.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416388\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836416408\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416422\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use definition of exponent.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416449\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416476\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829595190\" class=\"unnumbered unstyled\" summary=\"The expression is 3 raised to the power x. Replace x with 4 to get 3 to the power 4. Use definition of exponent to get 3 times 3 times 3 times 3. Simplify to get 81.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829595233\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829595253\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829595267\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use definition of exponent.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836556778\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836556805\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836556826\" class=\"unnumbered unstyled\" summary=\"The expression is 2 x squared plus 3 x plus 8. Substitute x with 4 to get 2 open parentheses 4 close parentheses squared plus 3 open parentheses 4 close parentheses plus 8. Follow order of operations to first get 2 open parentheses 16 close parentheses plus 3 open parentheses 4 close parentheses plus 8. Then, 32 plus 12 plus 8. Then, 52.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836556871\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836556890\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571162\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Follow the order of operations.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571189\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571214\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571240\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836571258\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836571262\"><div data-type=\"problem\" id=\"fs-id1167836571264\"><p id=\"fs-id1167836571266\">Evaluate when \\(x=3,\\) <span class=\"token\">\u24d0<\/span> \\({x}^{2}\\) <span class=\"token\">\u24d1<\/span> \\({4}^{x}\\) <span class=\"token\">\u24d2<\/span> \\(3{x}^{2}+4x+1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836627717\"><p id=\"fs-id1167836627719\"><span class=\"token\">\u24d0<\/span> 9 <span class=\"token\">\u24d1<\/span> 64 <span class=\"token\">\u24d2<\/span>40<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836627740\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836627744\"><div data-type=\"problem\" id=\"fs-id1167836627747\"><p id=\"fs-id1167836627749\">Evaluate when \\(x=6,\\) <span class=\"token\">\u24d0<\/span> \\({x}^{3}\\) <span class=\"token\">\u24d1<\/span> \\({2}^{x}\\) <span class=\"token\">\u24d2<\/span> \\(6{x}^{2}-4x-7.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836627820\"><p id=\"fs-id1167836627822\"><span class=\"token\">\u24d0<\/span> 216 <span class=\"token\">\u24d1<\/span> 64 <span class=\"token\">\u24d2<\/span> 185<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836627844\"><h3 data-type=\"title\">Identify and Combine Like Terms<\/h3><p id=\"fs-id1167836627849\">Algebraic expressions are made up of terms. A <span data-type=\"term\">term<\/span> is a constant, or the product of a constant and one or more variables.<\/p><div data-type=\"note\" id=\"fs-id1167836627856\"><div data-type=\"title\">Term<\/div><p id=\"fs-id1167836627861\">A <strong data-effect=\"bold\">term<\/strong> is a constant or the product of a constant and one or more variables.<\/p><\/div><p id=\"fs-id1167836448104\">Examples of terms are \\(7,y,5{x}^{2},9a,\\) and \\({b}^{5}.\\)<\/p><p id=\"fs-id1167836448144\">The constant that multiplies the variable is called the <span data-type=\"term\">coefficient<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836448151\"><div data-type=\"title\">Coefficient<\/div><p id=\"fs-id1167836448156\">The <strong data-effect=\"bold\">coefficient<\/strong> of a term is the constant that multiplies the variable in a term.<\/p><\/div><p id=\"fs-id1167836448165\">Think of the coefficient as the number in front of the variable. The coefficient of the term \\(3x\\) is 3. When we write \\(x,\\) the coefficient is 1, since \\(x=1\u00b7x.\\)<\/p><p id=\"fs-id1167836448199\">Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are <span data-type=\"term\">like terms<\/span>.<\/p><p id=\"fs-id1167836448207\">Look at the following 6 terms. Which ones seem to have traits in common?<\/p><div data-type=\"equation\" id=\"fs-id1167836448210\" class=\"unnumbered\" data-label=\"\">\\(5x\\phantom{\\rule{1em}{0ex}}7\\phantom{\\rule{1em}{0ex}}{n}^{2}\\phantom{\\rule{1em}{0ex}}4\\phantom{\\rule{1em}{0ex}}3x\\phantom{\\rule{1em}{0ex}}9{n}^{2}\\)<\/div><p id=\"fs-id1167836448258\">We say,<\/p><p id=\"fs-id1167836448262\">\\(\\phantom{\\rule{1em}{0ex}}7\\) and \\(4\\) are like terms.<\/p><p id=\"fs-id1167836652453\">\\(\\phantom{\\rule{1em}{0ex}}5x\\) and \\(3x\\) are like terms.<\/p><p id=\"fs-id1167836652473\">\\(\\phantom{\\rule{1em}{0ex}}{n}^{2}\\) and \\(9{n}^{2}\\) are like terms.<\/p><div data-type=\"note\" id=\"fs-id1167836652497\"><div data-type=\"title\">Like Terms<\/div><p id=\"fs-id1167836652502\">Terms that are either constants or have the same variables raised to the same powers are called <strong data-effect=\"bold\">like terms.<\/strong><\/p><\/div><p id=\"fs-id1167836652512\">If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.<\/p><div data-type=\"equation\" id=\"fs-id1167836652516\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}4x+7x+x\\hfill \\\\ \\text{Add the coefficients.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}12x\\hfill \\end{array}\\)<\/div><div data-type=\"example\" id=\"fs-id1167836652573\" class=\"textbox textbox--examples\"><div data-type=\"title\">How To Combine Like Terms<\/div><div data-type=\"exercise\" id=\"fs-id1167836652578\"><div data-type=\"problem\" id=\"fs-id1167836652581\"><p id=\"fs-id1167836652583\">Simplify: \\(2{x}^{2}+3x+7+{x}^{2}+4x+5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836611286\"><span data-type=\"media\" id=\"fs-id1167836611288\" data-alt=\"Step 1 is to identify the like terms in 2 x squared plus 3 x plus 7 plus x squared plus 4 x plus 5. The like terms are 2 x squared and x squared, then 3 x and 4 x, then 7 and 5.\"><img src=\"CNX_IntAlg_Figure_01_01_014_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to identify the like terms in 2 x squared plus 3 x plus 7 plus x squared plus 4 x plus 5. The like terms are 2 x squared and x squared, then 3 x and 4 x, then 7 and 5.\"><\/span><span data-type=\"media\" id=\"fs-id1167836611301\" data-alt=\"Step 2 is to rearrange the expression so the like terms are together. Hence, we have 2 x squared plus x squared plus 3 x plus 4 x plus 7 plus 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_014b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rearrange the expression so the like terms are together. Hence, we have 2 x squared plus x squared plus 3 x plus 4 x plus 7 plus 5.\"><\/span><span data-type=\"media\" id=\"fs-id1167836611314\" data-alt=\"Step 3 is to combine the like terms to get 3 x squared plus 7 x plus 12.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_014c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to combine the like terms to get 3 x squared plus 7 x plus 12.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836611328\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836611332\"><div data-type=\"problem\" id=\"fs-id1167836611335\"><p id=\"fs-id1167836611337\">Simplify: \\(3{x}^{2}+7x+9+7{x}^{2}+9x+8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836611384\"><p id=\"fs-id1167836611386\">\\(10{x}^{2}+16x+17\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836611411\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836611416\"><div data-type=\"problem\" id=\"fs-id1167836611418\"><p id=\"fs-id1167836611420\">Simplify: \\(4{y}^{2}+5y+2+8{y}^{2}+4y+5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836792868\"><p id=\"fs-id1167836792871\">\\(12{y}^{2}+9y+7\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836792896\" class=\"howto\"><div data-type=\"title\">Combine like terms.<\/div><ol id=\"fs-id1167836792904\" type=\"1\" class=\"stepwise\"><li>Identify like terms.<\/li><li>Rearrange the expression so like terms are together.<\/li><li>Add or subtract the coefficients and keep the same variable for each group of like terms.<\/li><\/ol><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836792925\"><h3 data-type=\"title\">Translate an English Phrase to an Algebraic Expression<\/h3><p id=\"fs-id1167836792931\">We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we\u2019ve talked about will help us do that. <a href=\"#fs-id1167836792941\" class=\"autogenerated-content\">(Figure)<\/a> summarizes them.<\/p><table id=\"fs-id1167836792941\" summary=\"This table has three columns labeled operation, phrase and expression. There are four rows. The phrases for addition are a plus b, the sum of a and b, a increased by b, the total of a and b, b added to a. The expression is a plus b. The phrases for subtraction are a minus b, the difference of a and b, a decreased by b, b less than a, b subtracted from a. The expression is a minus b. The phrases for multiplication are a times b, the product of a and b, 2a. The expressions are a dot b, ab, a open parentheses b close parentheses, open parentheses a parentheses open parentheses b close parentheses and 2a. The phrases for division are a divided by b, the quotient of a and b, the ratio of a and b, b divided into a. The expressions are a divided by b, a slash b, a upon b, b parentheses a overbar.\"><thead><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"left\">Operation<\/th><th data-valign=\"middle\" data-align=\"left\">Phrase<\/th><th data-valign=\"middle\" data-align=\"left\">Expression<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Addition<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> plus <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div>the sum of \\(a\\) and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">a<\/em> increased by <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> more than <em data-effect=\"italics\">a<\/em><div data-type=\"newline\"><br><\/div>the total of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> added to <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a+b\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Subtraction<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> minus \\(b\\)<div data-type=\"newline\"><br><\/div>the difference of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">a<\/em> decreased by <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> less than <em data-effect=\"italics\">a<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> subtracted from <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a-b\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Multiplication<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> times <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div>the product of \\(a\\) and \\(b\\)<div data-type=\"newline\"><br><\/div>twice <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a\u00b7b,ab,a\\left(b\\right),\\left(a\\right)\\left(b\\right)\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(2a\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Division<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> divided by <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div>the quotient of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div>the ratio of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> divided into <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a\u00f7b,a\\text{\/}b,\\frac{a}{b},b\\overline{)a}\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836513163\">Look closely at these phrases using the four operations:<\/p><span data-type=\"media\" id=\"fs-id1167836513166\" data-alt=\"The sum of a and b, the difference of a and b, the product of a and b, the quotient of a and b.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_015_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The sum of a and b, the difference of a and b, the product of a and b, the quotient of a and b.\"><\/span><p id=\"fs-id1167836513178\">Each phrase tells us to operate on two numbers. Look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to find the numbers.<\/p><div data-type=\"example\" id=\"fs-id1167836513192\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836513194\"><div data-type=\"problem\" id=\"fs-id1167836513196\"><p id=\"fs-id1167836513198\">Translate each English phrase into an algebraic expression:<\/p><p id=\"fs-id1167836513201\"><span class=\"token\">\u24d0<\/span> the difference of \\(14x\\) and 9<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the quotient of \\(8{y}^{2}\\) and 3<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> twelve more than \\(y\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> seven less than \\(49{x}^{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836520411\"><p id=\"fs-id1167836520413\"><span class=\"token\">\u24d0<\/span> The key word is <em data-effect=\"italics\">difference<\/em>, which tells us the operation is subtraction. Look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and t<\/em>o find the numbers to subtract.<\/p><span data-type=\"media\" id=\"fs-id1167836520436\" data-alt=\"The difference of 14 x and 9, 14 x minus 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_016_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The difference of 14 x and 9, 14 x minus 9.\"><\/span><p id=\"fs-id1167836520448\"><span class=\"token\">\u24d1<\/span> The key word is <em data-effect=\"italics\">quotient<\/em>, which tells us the operation is division.<\/p><span data-type=\"media\" id=\"fs-id1167836520460\" data-alt=\"The quotient of 8 y squared and 3, divide 8 y squared by 3, 8 y squared divided by 3. This can also be written as 8 y squared slash 3 or 8 y squared upon 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_017_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The quotient of 8 y squared and 3, divide 8 y squared by 3, 8 y squared divided by 3. This can also be written as 8 y squared slash 3 or 8 y squared upon 3.\"><\/span><p id=\"fs-id1167836520473\"><span class=\"token\">\u24d2<\/span> The key words are <em data-effect=\"italics\">more than.<\/em> They tell us the operation is addition. <em data-effect=\"italics\">More than<\/em> means \u201cadded to.\u201d<\/p><div data-type=\"equation\" id=\"fs-id1167836520491\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill \\text{twelve more than}\\phantom{\\rule{0.2em}{0ex}}y\\hfill \\\\ \\hfill \\text{twelve added to}\\phantom{\\rule{0.2em}{0ex}}y\\hfill \\\\ \\hfill y+12\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836520533\"><span class=\"token\">\u24d3<\/span> The key words are <em data-effect=\"italics\">less than<\/em>. They tell us to subtract. <em data-effect=\"italics\">Less than<\/em> means \u201csubtracted from.\u201d<\/p><div data-type=\"equation\" id=\"fs-id1167836520550\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill \\text{seven less than}\\phantom{\\rule{0.2em}{0ex}}49{x}^{2}\\hfill \\\\ \\hfill \\text{seven subtracted from}\\phantom{\\rule{0.2em}{0ex}}49{x}^{2}\\hfill \\\\ \\hfill 49{x}^{2}-7\\hfill \\end{array}\\)<\/div><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836520235\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836520239\"><div data-type=\"problem\" id=\"fs-id1167836520241\"><p id=\"fs-id1167836520243\">Translate the English phrase into an algebraic expression:<\/p><p id=\"fs-id1167836520246\"><span class=\"token\">\u24d0<\/span> the difference of \\(14{x}^{2}\\) and 13<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the quotient of \\(12x\\) and 2<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> 13 more than \\(z\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> 18 less than \\(8x\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836520301\"><p id=\"fs-id1167836520303\"><span class=\"token\">\u24d0<\/span>\\(14{x}^{2}-13\\)<span class=\"token\">\u24d1<\/span>\\(12x\u00f72\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(z+13\\)<span class=\"token\">\u24d3<\/span>\\(8x-18\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836513469\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836513472\"><div data-type=\"problem\" id=\"fs-id1167836513474\"><p id=\"fs-id1167836513476\">Translate the English phrase into an algebraic expression:<\/p><p id=\"fs-id1167836513480\"><span class=\"token\">\u24d0<\/span> the sum of \\(17{y}^{2}\\) and 19<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the product of \\(7\\) and <em data-effect=\"italics\">y<\/em><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Eleven more than <em data-effect=\"italics\">x<\/em><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> Fourteen less than 11<em data-effect=\"italics\">a<\/em><\/div><div data-type=\"solution\" id=\"fs-id1167836513534\"><p id=\"fs-id1167836513536\"><span class=\"token\">\u24d0<\/span>\\(17{y}^{2}+19\\)<span class=\"token\">\u24d1<\/span>\\(7y\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(x+11\\)<span class=\"token\">\u24d3<\/span>\\(11a-14\\)<\/div><\/div><\/div><p id=\"fs-id1171790448010\">We look carefully at the words to help us distinguish between multiplying a sum and adding a product.<\/p><div data-type=\"example\" id=\"fs-id1167836513605\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836513607\"><div data-type=\"problem\" id=\"fs-id1167836513609\"><p id=\"fs-id1167836513612\">Translate the English phrase into an algebraic expression:<\/p><p id=\"fs-id1167836513615\"><span class=\"token\">\u24d0<\/span> eight times the sum of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the sum of eight times <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em><\/div><div data-type=\"solution\" id=\"fs-id1167836512532\"><p id=\"fs-id1167836512534\">There are two operation words\u2014<em data-effect=\"italics\">times<\/em> tells us to multiply and <em data-effect=\"italics\">sum<\/em> tells us to add.<\/p><p id=\"fs-id1167836512547\"><span class=\"token\">\u24d0<\/span> Because we are multiplying 8 times the sum, we need parentheses around the sum of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, \\(\\left(x+y\\right).\\) This forces us to determine the sum first. (Remember the order of operations.)<\/p><div data-type=\"equation\" id=\"fs-id1167836512583\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill \\text{eight times the sum of}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}y\\hfill \\\\ \\hfill 8\\left(x+y\\right)\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836512634\"><span class=\"token\">\u24d1<\/span> To take a sum, we look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to see what is being added. Here we are taking the sum <em data-effect=\"italics\">of<\/em> eight times <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/p><span data-type=\"media\" id=\"fs-id1167836512668\" data-alt=\"The sum of 8 times x and y is 8 x plus y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_018_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The sum of 8 times x and y is 8 x plus y.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836620952\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836620957\"><div data-type=\"problem\" id=\"fs-id1167836620959\"><p id=\"fs-id1167836620961\">Translate the English phrase into an algebraic expression:<\/p><p id=\"fs-id1167836620964\"><span class=\"token\">\u24d0<\/span> four times the sum of <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the sum of four times <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em><\/div><div data-type=\"solution\" id=\"fs-id1167836620997\"><p id=\"fs-id1167836621000\"><span class=\"token\">\u24d0<\/span>\\(4\\left(p+q\\right)\\)<span class=\"token\">\u24d1<\/span>\\(4p+q\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836621044\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836621048\"><div data-type=\"problem\" id=\"fs-id1167836621050\"><p id=\"fs-id1167836621052\">Translate the English phrase into an algebraic expression:<\/p><p id=\"fs-id1167836621056\"><span class=\"token\">\u24d0<\/span> the difference of two times <em data-effect=\"italics\">x<\/em> and 8<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> two times the difference of <em data-effect=\"italics\">x<\/em> and 8<\/div><div data-type=\"solution\" id=\"fs-id1167836621081\"><p id=\"fs-id1167836621083\"><span class=\"token\">\u24d0<\/span>\\(2x-8\\)<span class=\"token\">\u24d1<\/span>\\(2\\left(x-8\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836606926\">Later in this course, we\u2019ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We\u2019ll see how to do this in the next two examples.<\/p><div data-type=\"example\" id=\"fs-id1167836606933\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836606935\"><div data-type=\"problem\" id=\"fs-id1167836606937\"><p id=\"fs-id1167836606939\">The length of a rectangle is 14 less than the width. Let <em data-effect=\"italics\">w<\/em> represent the width of the rectangle. Write an expression for the length of the rectangle.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836606950\"><p id=\"fs-id1167836606952\">\\(\\begin{array}{cccc}\\text{Write a phrase about the length of the rectangle.}\\phantom{\\rule{5em}{0ex}}\\hfill &amp; &amp; &amp; \\hfill \\text{14 less than the width}\\hfill \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}w\\phantom{\\rule{0.2em}{0ex}}\\text{for \u201cthe width.\u201d}\\hfill &amp; &amp; &amp; \\hfill w\\hfill \\\\ \\text{Rewrite}\\phantom{\\rule{0.2em}{0ex}}{\\text{less than}}\\phantom{\\rule{0.2em}{0ex}}\\text{as}\\phantom{\\rule{0.2em}{0ex}}{\\text{subtracted from}}.\\hfill &amp; &amp; &amp; \\hfill \\text{14 subtracted from}\\phantom{\\rule{0.2em}{0ex}}w\\hfill \\\\ \\text{Translate the phrase into algebra.}\\hfill &amp; &amp; &amp; \\hfill w-14\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836607056\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836607060\"><div data-type=\"problem\" id=\"fs-id1167836607063\"><p id=\"fs-id1167836607065\">The length of a rectangle is 7 less than the width. Let <em data-effect=\"italics\">w<\/em> represent the width of the rectangle. Write an expression for the length of the rectangle.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836607075\"><p id=\"fs-id1167836607077\">\\(w-7\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829696780\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829696784\"><div data-type=\"problem\" id=\"fs-id1167829696786\"><p id=\"fs-id1167829696788\">The width of a rectangle is 6 less than the length. Let <em data-effect=\"italics\">l<\/em> represent the length of the rectangle. Write an expression for the width of the rectangle.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829696798\"><p id=\"fs-id1167829696801\">\\(l-6\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1171790738586\">The expressions in the next example will be used in the typical coin mixture problems we will see soon.<\/p><div data-type=\"example\" id=\"fs-id1167829696814\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829696817\"><div data-type=\"problem\" id=\"fs-id1167829696819\"><p id=\"fs-id1167829696821\">June has dimes and quarters in her purse. The number of dimes is seven less than four times the number of quarters. Let <em data-effect=\"italics\">q<\/em> represent the number of quarters. Write an expression for the number of dimes.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829696832\"><p id=\"fs-id1167829696834\">\\(\\begin{array}{cccc}\\text{Write a phrase about the number of dimes.}\\hfill &amp; &amp; &amp; \\hfill \\text{seven less than four times the number of quarters}\\hfill \\\\ \\\\ \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}q\\phantom{\\rule{0.2em}{0ex}}\\text{for the number of quarters.}\\hfill &amp; &amp; &amp; \\hfill \\text{7 less than 4 times}\\phantom{\\rule{0.2em}{0ex}}q\\hfill \\\\ \\text{Translate 4 times}\\phantom{\\rule{0.2em}{0ex}}q.\\hfill &amp; &amp; &amp; \\hfill \\text{7 less than 4}q\\hfill \\\\ \\text{Translate the phrase into algebra.}\\hfill &amp; &amp; &amp; \\hfill 4q-7\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829696930\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829696934\"><div data-type=\"problem\" id=\"fs-id1167829696936\"><p id=\"fs-id1167829696939\">Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let <em data-effect=\"italics\">q<\/em> represent the number of quarters. Write an expression for the number of dimes.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836511104\"><p id=\"fs-id1167836511106\">\\(4q-8\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836511122\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836511126\"><div data-type=\"problem\" id=\"fs-id1167836511128\"><p id=\"fs-id1167836511130\">Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let <em data-effect=\"italics\">n<\/em> represent the number of nickels. Write an expression for the number of dimes.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836511141\"><p id=\"fs-id1167836511143\">\\(7n+3\\)<\/p><\/div><\/div><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836511160\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836511167\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Divisibility Tests<\/strong><div data-type=\"newline\"><br><\/div> A number is divisible by:<div data-type=\"newline\"><br><\/div> \u2003\u20032 if the last digit is 0, 2, 4, 6, or 8.<div data-type=\"newline\"><br><\/div> \u2003\u20033 if the sum of the digits is divisible by 3.<div data-type=\"newline\"><br><\/div> \u2003\u20035 if the last digit is 5 or 0.<div data-type=\"newline\"><br><\/div> \u2003\u20036 if it is divisible by both 2 and 3.<div data-type=\"newline\"><br><\/div> \u2003\u200310 if it ends with 0.<\/li><li><strong data-effect=\"bold\">How to find the prime factorization of a composite number.<\/strong><div data-type=\"newline\"><br><\/div><ol id=\"fs-id1167836511200\" type=\"1\" class=\"stepwise\"><li>Find two factors whose product is the given number, and use these numbers to create two branches.<\/li><li>If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.<\/li><li>If a factor is not prime, write it as the product of two factors and continue the process.<\/li><li>Write the composite number as the product of all the circled primes.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How To Find the least common multiple using the prime factors method.<\/strong><div data-type=\"newline\"><br><\/div><ol id=\"fs-id1167836511232\" type=\"1\" class=\"stepwise\"><li>Write each number as a product of primes.<\/li><li>List the primes of each number. Match primes vertically when possible.<\/li><li>Bring down the columns.<\/li><li>Multiply the factors.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Equality Symbol<\/strong><div data-type=\"newline\"><br><\/div>\\(a=b\\) is read \u201c<em data-effect=\"italics\">a<\/em> is equal to <em data-effect=\"italics\">b<\/em>.\u201d<div data-type=\"newline\"><br><\/div> The symbol \u201c=\u201d is called the equal sign.<\/li><li><strong data-effect=\"bold\">Inequality<\/strong><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831871677\" data-alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_020_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\"><\/span><\/li><li><strong data-effect=\"bold\">Inequality Symbols<\/strong><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836519850\" summary=\"The table describes inequality symbols in words. The symbols described are a is not equal to b, a is less than b, a is less than or equal to b, a is greater then b, a is greater than or equal to b.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Inequality Symbols<\/th><th data-valign=\"top\" data-align=\"left\">Words<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a\\ne b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">not equal to b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a&lt;b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a\\le b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than or equal to b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a&gt;b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than b.<\/em><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\">\\(a\\ge b\\)<\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than or equal to b.<\/em><\/td><\/tr><\/tbody><\/table><\/li><li><strong data-effect=\"bold\">Grouping Symbols<\/strong><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}\\text{Parentheses}\\hfill &amp; &amp; &amp; &amp; &amp; \\left(\\phantom{\\rule{0.2em}{0ex}}\\right)\\hfill \\\\ \\text{Brackets}\\hfill &amp; &amp; &amp; &amp; &amp; \\left[\\phantom{\\rule{0.2em}{0ex}}\\right]\\hfill \\\\ \\text{Braces}\\hfill &amp; &amp; &amp; &amp; &amp; \\left\\{\\phantom{\\rule{0.2em}{0ex}}\\right\\}\\hfill \\end{array}\\)<\/li><li><strong data-effect=\"bold\">Exponential Notation<\/strong><div data-type=\"newline\"><br><\/div>\\({a}^{n}\\) means multiply <em data-effect=\"italics\">a<\/em> by itself, <em data-effect=\"italics\">n<\/em> times.<div data-type=\"newline\"><br><\/div> The expression \\({a}^{n}\\) is read <em data-effect=\"italics\">a<\/em> to the \\({n}^{th}\\) power.<\/li><li><strong data-effect=\"bold\">Simplify an Expression<\/strong><div data-type=\"newline\"><br><\/div> To simplify an expression, do all operations in the expression.<\/li><li><strong data-effect=\"bold\">How to use the order of operations.<\/strong><div data-type=\"newline\"><br><\/div><ol id=\"fs-id1167836545484\" type=\"1\" class=\"stepwise\"><li>Parentheses and Other Grouping Symbols <ul id=\"fs-id1167836545494\" data-bullet-style=\"bullet\"><li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li><\/ul><\/li><li>Exponents <ul id=\"fs-id1167836545508\" data-bullet-style=\"bullet\"><li>Simplify all expressions with exponents.<\/li><\/ul><\/li><li>Multiplication and Division <ul id=\"fs-id1167836545520\" data-bullet-style=\"bullet\"><li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li><\/ul><\/li><li>Addition and Subtraction <ul id=\"fs-id1167836545534\" data-bullet-style=\"bullet\"><li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li><\/ul><\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to combine like terms.<\/strong><div data-type=\"newline\"><br><\/div><ol id=\"fs-id1167836545552\" type=\"1\" class=\"stepwise\"><li>Identify like terms.<\/li><li>Rearrange the expression so like terms are together.<\/li><li>Add or subtract the coefficients and keep the same variable for each group of like terms.<\/li><\/ol><table id=\"fs-id1167836545572\" summary=\"This table has three columns labeled operation, phrase and expression. There are four rows. The phrases for addition are a plus b, the sum of a and b, a increased by b, the total of a and b, b added to a. The expression is a plus b. The phrases for subtraction are a minus b, the difference of a and b, a decreased by b, b less than a, b subtracted from a. The expression is a minus b. The phrases for multiplication are a times b, the product of a and b, 2a. The expressions are a dot b, ab, a open parentheses b close parentheses, open parentheses a parentheses open parentheses b close parentheses and 2a. The phrases for division are a divided by b, the quotient of a and b, the ratio of a and b, b divided into a. The expressions are a divided by b, a slash b, a upon b, b parentheses a overbar.\"><thead><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"left\">Operation<\/th><th data-valign=\"middle\" data-align=\"left\">Phrase<\/th><th data-valign=\"middle\" data-align=\"left\">Expression<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Addition<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> plus <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div>the sum of \\(a\\) and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">a<\/em> increased by <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> more than <em data-effect=\"italics\">a<\/em><div data-type=\"newline\"><br><\/div>the total of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> added to <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a+b\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Subtraction<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> minus \\(b\\)<div data-type=\"newline\"><br><\/div>the difference of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">a<\/em> decreased by <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> less than <em data-effect=\"italics\">a<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> subtracted from <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a-b\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Multiplication<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> times <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div>the product of \\(a\\) and \\(b\\)<div data-type=\"newline\"><br><\/div>twice <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a\u00b7b,ab,a\\left(b\\right),\\left(a\\right)\\left(b\\right)\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(2a\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Division<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> divided by <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>the quotient of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div>the ratio of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">b<\/em> divided into <em data-effect=\"italics\">a<\/em><\/td><td data-valign=\"top\" data-align=\"left\">\\(a\u00f7b,a\\text{\/}b,\\frac{a}{b},b\\overline{)a}\\)<\/td><\/tr><\/tbody><\/table><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829693720\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829693724\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167829693732\"><strong data-effect=\"bold\">Identify Multiples and Factors<\/strong><\/p><p id=\"fs-id1167829693738\">In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.<\/p><div data-type=\"exercise\" id=\"fs-id1167829693742\"><div data-type=\"problem\" id=\"fs-id1167829693745\"><p id=\"fs-id1167829693747\">84<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829693751\"><p id=\"fs-id1167829693753\">Divisible by 2, 3, 6<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829693758\"><div data-type=\"problem\" id=\"fs-id1167829693761\"><p id=\"fs-id1167829693763\">96<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829693774\"><div data-type=\"problem\" id=\"fs-id1167829693777\"><p id=\"fs-id1167829693779\">896<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829693783\"><p id=\"fs-id1167829693785\">Divisible by 2<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829693790\"><div data-type=\"problem\" id=\"fs-id1167829693793\"><p id=\"fs-id1167829693795\">942<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706817\"><div data-type=\"problem\" id=\"fs-id1167836706819\"><p id=\"fs-id1167836706821\">22,335<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836706825\"><p id=\"fs-id1167836706828\">Divisible by 3, 5<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706833\"><div data-type=\"problem\" id=\"fs-id1167836706835\"><p id=\"fs-id1167836706837\">39,075<\/p><\/div><\/div><p id=\"fs-id1167836706849\"><strong data-effect=\"bold\">Find Prime Factorizations and Least Common Multiples<\/strong><\/p><p id=\"fs-id1167836706854\">In the following exercises, find the prime factorization.<\/p><div data-type=\"exercise\" id=\"fs-id1167836706857\"><div data-type=\"problem\" id=\"fs-id1167836706860\"><p id=\"fs-id1167836706862\">86<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836706866\"><p id=\"fs-id1167836706868\">\\(2\u00b743\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706881\"><div data-type=\"problem\" id=\"fs-id1167836706883\"><p id=\"fs-id1167836706885\">78<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706909\"><div data-type=\"problem\" id=\"fs-id1167836706911\"><p id=\"fs-id1167836706913\">455<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836706917\"><p id=\"fs-id1167836706919\">\\(5\u00b77\u00b713\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706936\"><div data-type=\"problem\" id=\"fs-id1167836706938\"><p id=\"fs-id1167836706941\">400<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706977\"><div data-type=\"problem\" id=\"fs-id1167836706979\"><p id=\"fs-id1167836706981\">432<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836706985\"><p id=\"fs-id1167836706988\">\\(2\u00b72\u00b72\u00b72\u00b73\u00b73\u00b73\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836579693\"><div data-type=\"problem\" id=\"fs-id1167836579695\"><p id=\"fs-id1167836579697\">627<\/p><\/div><\/div><p id=\"fs-id1167836579721\">In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.<\/p><div data-type=\"exercise\" id=\"fs-id1167836579725\"><div data-type=\"problem\" id=\"fs-id1167836579727\"><p id=\"fs-id1167836579729\">8, 12<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836579734\"><p id=\"fs-id1167836579736\">24<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836579741\"><div data-type=\"problem\" id=\"fs-id1167836579743\"><p id=\"fs-id1167836579745\">12, 16<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836579757\"><div data-type=\"problem\" id=\"fs-id1167836579759\"><p id=\"fs-id1167836579761\">28, 40<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836579766\"><p id=\"fs-id1167836579768\">420<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836579773\"><div data-type=\"problem\" id=\"fs-id1167836579775\"><p id=\"fs-id1167836579777\">84, 90<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836579789\"><div data-type=\"problem\" id=\"fs-id1167836579791\"><p id=\"fs-id1167836579793\">55, 88<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836579798\"><p id=\"fs-id1167836579800\">440<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836579805\"><div data-type=\"problem\" id=\"fs-id1167836579807\"><p id=\"fs-id1167836579809\">60, 72<\/p><\/div><\/div><p id=\"fs-id1167836579821\"><strong data-effect=\"bold\">Simplify Expressions Using the Order of Operations<\/strong><\/p><p id=\"fs-id1167836579827\">In the following exercises, simplify each expression.<\/p><div data-type=\"exercise\" id=\"fs-id1167836579831\"><div data-type=\"problem\" id=\"fs-id1167836579833\"><p id=\"fs-id1167836579835\">\\({2}^{3}-12\u00f7\\left(9-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836692441\"><p id=\"fs-id1167836692443\">5<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836692448\"><div data-type=\"problem\" id=\"fs-id1167836692450\"><p id=\"fs-id1167836692452\">\\({3}^{2}-18\u00f7\\left(11-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836692492\"><div data-type=\"problem\" id=\"fs-id1167836692494\"><p id=\"fs-id1167836692496\">\\(2+8\\left(6+1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836692522\"><p id=\"fs-id1167836692524\">58<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836692530\"><div data-type=\"problem\" id=\"fs-id1167836692532\"><p id=\"fs-id1167836692534\">\\(4+6\\left(3+6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836692568\"><div data-type=\"problem\" id=\"fs-id1167836692570\"><p id=\"fs-id1167836692572\">\\(20\u00f74+6\\left(5-1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836620005\"><p id=\"fs-id1167836620007\">29<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836620012\"><div data-type=\"problem\" id=\"fs-id1167836620015\"><p id=\"fs-id1167836620017\">\\(33\u00f73+4\\left(7-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836620053\"><div data-type=\"problem\" id=\"fs-id1167836620055\"><p id=\"fs-id1167836620057\">\\(3\\left(1+9\u00b76\\right)-{4}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836620091\"><p id=\"fs-id1167836620093\">149<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836620099\"><div data-type=\"problem\" id=\"fs-id1167836620101\"><p id=\"fs-id1167836620103\">\\(5\\left(2+8\u00b74\\right)-{7}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836620144\"><div data-type=\"problem\" id=\"fs-id1167836620146\"><p id=\"fs-id1167836620148\">\\(2\\left[1+3\\left(10-2\\right)\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836620182\"><p id=\"fs-id1167836620184\">50<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836685118\"><div data-type=\"problem\" id=\"fs-id1167836685120\"><p id=\"fs-id1167836685122\">\\(5\\left[2+4\\left(3-2\\right)\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836685162\"><div data-type=\"problem\" id=\"fs-id1167836685164\"><p id=\"fs-id1167836685167\">\\(8+2\\left[7-2\\left(5-3\\right)\\right]-{3}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836685211\"><p id=\"fs-id1167836685214\">5<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836685219\"><div data-type=\"problem\" id=\"fs-id1167836685221\"><p id=\"fs-id1167836685223\">\\(10+3\\left[6-2\\left(4-2\\right)\\right]-{2}^{4}\\)<\/p><\/div><\/div><p id=\"fs-id1167836685275\"><strong data-effect=\"bold\">Evaluate an Expression<\/strong><\/p><p id=\"fs-id1167836685282\">In the following exercises, evaluate the following expressions.<\/p><div data-type=\"exercise\" id=\"fs-id1167836685285\"><div data-type=\"problem\" id=\"fs-id1167836685287\"><p id=\"fs-id1167836685289\">When \\(x=2,\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\({x}^{6}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\({4}^{x}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(2{x}^{2}+3x-7\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833048379\"><p id=\"fs-id1167833048381\"><span class=\"token\">\u24d0<\/span> 64 <span class=\"token\">\u24d1<\/span> 16 <span class=\"token\">\u24d2<\/span> 7<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833048401\"><div data-type=\"problem\" id=\"fs-id1167833048403\"><p id=\"fs-id1167833048405\">When \\(x=3,\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\({x}^{5}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\({5}^{x}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(3{x}^{2}-4x-8\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833048496\"><div data-type=\"problem\" id=\"fs-id1167833048498\"><p id=\"fs-id1167833048500\">When \\(x=4,y=1\\)<\/p><div data-type=\"newline\"><br><\/div>\\({x}^{2}+3xy-7{y}^{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836526227\"><p id=\"fs-id1167836526229\">21<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836526235\"><div data-type=\"problem\" id=\"fs-id1167836526237\"><p id=\"fs-id1167836526239\">When \\(x=3,y=2\\)<\/p><div data-type=\"newline\"><br><\/div>\\(6{x}^{2}+3xy-9{y}^{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836526295\"><div data-type=\"problem\" id=\"fs-id1167836526298\"><p id=\"fs-id1167836526300\">When \\(x=10,y=7\\)<\/p><div data-type=\"newline\"><br><\/div>\\({\\left(x-y\\right)}^{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836526342\"><p id=\"fs-id1167836526344\">9<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836526350\"><div data-type=\"problem\" id=\"fs-id1167836526352\"><p id=\"fs-id1167836526354\">When \\(a=3,b=8\\)<\/p><div data-type=\"newline\"><br><\/div>\\({a}^{2}+{b}^{2}\\)<\/div><\/div><p id=\"fs-id1167836543680\"><strong data-effect=\"bold\">Simplify Expressions by Combining Like Terms<\/strong><\/p><p id=\"fs-id1167836543686\">In the following exercises, simplify the following expressions by combining like terms.<\/p><div data-type=\"exercise\" id=\"fs-id1167836543689\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836543691\"><p id=\"fs-id1167836543694\">\\(7x+2+3x+4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836543718\"><p id=\"fs-id1167836543720\">\\(10x+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836543735\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836543737\"><p id=\"fs-id1167836543739\">\\(8y+5+2y-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836543781\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836543783\"><p id=\"fs-id1167836543785\">\\(10a+7+5a-2+7a-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836543820\"><p id=\"fs-id1167836543823\">\\(22a+1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688236\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836688238\"><p id=\"fs-id1167836688240\">\\(7c+4+6c-3+9c-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688288\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836688290\"><p id=\"fs-id1167836688292\">\\(3{x}^{2}+12x+11+14{x}^{2}+8x+5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836688336\"><p id=\"fs-id1167836688338\">\\(17{x}^{2}+20x+16\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688363\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836688365\"><p id=\"fs-id1167836688367\">\\(5{b}^{2}+9b+10+2{b}^{2}+3b-4\\)<\/p><\/div><\/div><p id=\"fs-id1167836536429\"><strong data-effect=\"bold\">Translate an English Phrase to an Algebraic Expression<\/strong><\/p><p id=\"fs-id1167836536435\">In the following exercises, translate the phrases into algebraic expressions.<\/p><div data-type=\"exercise\" id=\"fs-id1167836536439\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836536441\"><p id=\"fs-id1167836536443\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> the difference of \\(5{x}^{2}\\) and \\(6xy\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the quotient of \\(6{y}^{2}\\) and \\(5x\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Twenty-one more than \\({y}^{2}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> \\(6x\\) less than \\(81{x}^{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836536531\"><p id=\"fs-id1167836536533\"><span class=\"token\">\u24d0<\/span>\\(5{x}^{2}-6xy\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{6{y}^{2}}{5x}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({y}^{2}+21\\)<span class=\"token\">\u24d3<\/span>\\(81{x}^{2}-6x\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836558971\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836558973\"><p id=\"fs-id1167836558975\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> the difference of \\(17{x}^{2}\\) and \\(5xy\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the quotient of \\(8{y}^{3}\\) and \\(3x\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Eighteen more than \\({a}^{2}\\);<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> \\(11b\\) less than \\(100{b}^{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833329552\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833329554\"><p id=\"fs-id1167833329556\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> the sum of \\(4a{b}^{2}\\) and \\(3{a}^{2}b\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the product of \\(4{y}^{2}\\) and \\(5x\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Fifteen more than \\(m\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> \\(9x\\) less than \\(121{x}^{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833329644\"><p id=\"fs-id1167833329646\"><span class=\"token\">\u24d0<\/span>\\(4a{b}^{2}+3{a}^{2}b\\)<span class=\"token\">\u24d1<\/span>\\(20x{y}^{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(m+15\\)<span class=\"token\">\u24d3<\/span>\\(121{x}^{2}-9x\\)\\(9x&lt;121{x}^{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833412878\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833412880\"><p id=\"fs-id1167833412882\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> the sum of \\(3{x}^{2}y\\) and \\(7x{y}^{2}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the product of \\(6x{y}^{2}\\) and \\(4z\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Twelve more than \\(3{x}^{2}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> \\(7{x}^{2}\\) less than \\(63{x}^{3}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832930223\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832930225\"><p id=\"fs-id1167832930227\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> eight times the difference of \\(y\\) and nine<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the difference of eight times \\(y\\) and 9<\/div><div data-type=\"solution\" id=\"fs-id1167832930249\"><p id=\"fs-id1167832930251\"><span class=\"token\">\u24d0<\/span>\\(8\\left(y-9\\right)\\)<span class=\"token\">\u24d1<\/span>\\(8y-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832930295\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832930297\"><p id=\"fs-id1167832930299\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> seven times the difference of \\(y\\) and one<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the difference of seven times \\(y\\) and 1<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832930367\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832930369\"><p id=\"fs-id1167832930371\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> five times the sum of \\(3x\\) and \\(y\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the sum of five times \\(3x\\) and \\(y\\)<\/div><div data-type=\"solution\" id=\"fs-id1167832951106\"><p id=\"fs-id1167832951108\"><span class=\"token\">\u24d0<\/span>\\(5\\left(3x+y\\right)\\)<span class=\"token\">\u24d1<\/span>\\(15x+y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832951153\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832951156\"><p id=\"fs-id1167832951158\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> eleven times the sum of \\(4{x}^{2}\\) and \\(5x\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> the sum of eleven times \\(4{x}^{2}\\) and \\(5x\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832951265\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832951267\"><p id=\"fs-id1167832951269\">Eric has rock and country songs on his playlist. The number of rock songs is 14 more than twice the number of country songs. Let <em data-effect=\"italics\">c<\/em> represent the number of country songs. Write an expression for the number of rock songs.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832951281\"><p id=\"fs-id1167832951283\">\\(14&gt;2c\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836518238\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836518240\"><p id=\"fs-id1167836518242\">The number of women in a Statistics class is 8 more than twice the number of men. Let \\(m\\) represent the number of men. Write an expression for the number of women.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836518268\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836518270\"><p id=\"fs-id1167836518272\">Greg has nickels and pennies in his pocket. The number of pennies is seven less than three the number of nickels. Let <em data-effect=\"italics\">n<\/em> represent the number of nickels. Write an expression for the number of pennies.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836518282\"><p id=\"fs-id1167836518284\">\\(3n-7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836518299\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836518302\"><p id=\"fs-id1167836518304\">Jeannette has \\(\\text{?}5\\) and \\(\\text{?}10\\) bills in her wallet. The number of fives is three more than six times the number of tens. Let \\(t\\) represent the number of tens. Write an expression for the number of fives.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836518346\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167836518354\"><div data-type=\"problem\" id=\"fs-id1167836518356\"><p id=\"fs-id1167836518358\">Explain in your own words how to find the prime factorization of a composite number.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836518362\"><p id=\"fs-id1167836518364\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836518370\"><div data-type=\"problem\" id=\"fs-id1167836518372\"><p id=\"fs-id1167836518374\">Why is it important to use the order of operations to simplify an expression?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836518386\"><div data-type=\"problem\" id=\"fs-id1167836518388\"><p id=\"fs-id1167836518390\">Explain how you identify the like terms in the expression \\(8{a}^{2}+4a+9-{a}^{2}-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836518428\"><p id=\"fs-id1167836518430\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836518436\"><div data-type=\"problem\" id=\"fs-id1167836518438\"><p id=\"fs-id1167836518440\">Explain the difference between the phrases \u201c4 times the sum of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>\u201d and \u201cthe sum of 4 times <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>\u201d.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836524988\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167836524994\"><span class=\"token\">\u24d0<\/span> Use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167836525008\" data-alt=\"This table has 4 columns, 7 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: identify multiples and apply divisibility tests, find prime factorizations and least common multiples, use variables and algebraic symbols, simplify expressions using the order of operations, evaluate an expression, identify and combine like terms, translate English phrases to algebraic expressions. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 7 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: identify multiples and apply divisibility tests, find prime factorizations and least common multiples, use variables and algebraic symbols, simplify expressions using the order of operations, evaluate an expression, identify and combine like terms, translate English phrases to algebraic expressions. The remaining columns are blank.\"><\/span><p id=\"fs-id1167836525017\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p><p id=\"fs-id1167836525024\"><strong data-effect=\"bold\">\u2026confidently.<\/strong> Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p><p id=\"fs-id1167836525034\"><strong data-effect=\"bold\">\u2026with some help.<\/strong> This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p><p id=\"fs-id1167836525045\"><strong data-effect=\"bold\">\u2026no - I don\u2019t get it!<\/strong> This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167836525060\"><dt>coefficient<\/dt><dd id=\"fs-id1167836525065\">The coefficient of a term is the constant that multiplies the variable in a term.<\/dd><\/dl><dl id=\"fs-id1167836525069\"><dt>composite number<\/dt><dd id=\"fs-id1167836525075\">A composite number is a counting number that is not prime. It has factors other than 1 and the number itself.<\/dd><\/dl><dl id=\"fs-id1167836525080\"><dt>constant<\/dt><dd id=\"fs-id1167836525085\">A constant is a number whose value always stays the same.<\/dd><\/dl><dl id=\"fs-id1167836525090\"><dt>divisible by a number<\/dt><dd id=\"fs-id1167836525095\">If a number <em data-effect=\"italics\">m<\/em> is a multiple of <em data-effect=\"italics\">n<\/em>, then <em data-effect=\"italics\">m<\/em> is divisible by <em data-effect=\"italics\">n<\/em>.<\/dd><\/dl><dl id=\"fs-id1167836525120\"><dt>equation<\/dt><dd id=\"fs-id1167836525125\">An equation is two expressions connected by an equal sign.<\/dd><\/dl><dl id=\"fs-id1167836525130\"><dt>evaluate an expression<\/dt><dd id=\"fs-id1167836525135\">To evaluate an expression means to find the value of the expression when the variables are replaced by a given number.<\/dd><\/dl><dl id=\"fs-id1167836525140\"><dt>expression<\/dt><dd id=\"fs-id1167836525146\">An expression is a number, a variable, or a combination of numbers and variables using operation symbols.<\/dd><\/dl><dl id=\"fs-id1167836525151\"><dt>factors<\/dt><dd id=\"fs-id1167836525156\">If \\(a\u00b7b=m,\\) then <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are factors of <em data-effect=\"italics\">m<\/em>.<\/dd><\/dl><dl id=\"fs-id1167833018659\"><dt>least common multiple<\/dt><dd id=\"fs-id1167833018664\">The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.<\/dd><\/dl><dl id=\"fs-id1167833018669\"><dt>like terms<\/dt><dd id=\"fs-id1167833018675\">Terms that are either constants or have the same variables raised to the same powers are called like terms.<\/dd><\/dl><dl id=\"fs-id1167833018680\"><dt>multiple of a number<\/dt><dd id=\"fs-id1167833018685\">A number is a multiple of <em data-effect=\"italics\">n<\/em> if it is the product of a counting number and <em data-effect=\"italics\">n.<\/em><\/dd><\/dl><dl id=\"fs-id1167833018699\"><dt>order of operations<\/dt><dd id=\"fs-id1167833018704\">The order of operations are established guidelines for simplifying an expression.<\/dd><\/dl><dl id=\"fs-id1167833018708\"><dt>prime factorization<\/dt><dd id=\"fs-id1167833018714\">The prime factorization of a number is the product of prime numbers that equals the number.<\/dd><\/dl><dl id=\"fs-id1167833018719\"><dt>prime number<\/dt><dd id=\"fs-id1167833018724\">A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.<\/dd><\/dl><dl id=\"fs-id1167833018730\"><dt>simplify an expression<\/dt><dd id=\"fs-id1167833018735\">To simplify an expression means to do all the math possible.<\/dd><\/dl><dl id=\"fs-id1167833018739\"><dt>term<\/dt><dd id=\"fs-id1167833018744\">A term is a constant, or the product of a constant and one or more variables.<\/dd><\/dl><dl id=\"fs-id1167833018749\"><dt>variable<\/dt><dd id=\"fs-id1167833018754\">A variable is a letter that represents a number whose value may change.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Find factors, prime factorizations, and least common multiples<\/li>\n<li>Use variables and algebraic symbols<\/li>\n<li>Simplify expressions using the order of operations<\/li>\n<li>Evaluate an expression<\/li>\n<li>Identify and combine like terms<\/li>\n<li>Translate an English phrase to an algebraic expression<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836546004\" class=\"be-prepared\">\n<p id=\"fs-id1167833056736\">This chapter is intended to be a brief review of concepts that will be needed in an Intermediate Algebra course. A more thorough introduction to the topics covered in this chapter can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833061637\">\n<h3 data-type=\"title\">Find Factors, Prime Factorizations, and Least Common Multiples<\/h3>\n<p id=\"fs-id1167836334858\">The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A <span data-type=\"term\">multiple<\/span> of 2 can be written as the product of a counting number and 2.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836376102\" data-alt=\"Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.\" \/><\/span><\/p>\n<p id=\"fs-id1167836349436\">Similarly, a multiple of 3 would be the product of a counting number and 3.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836282492\" data-alt=\"Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.\" \/><\/span><\/p>\n<p id=\"fs-id1167836360898\">We could find the multiples of any number by continuing this process.<\/p>\n<table id=\"fs-id1167836545969\" class=\"unnumbered\" summary=\"This table has 13 columns, 8 rows and a header row. The header row labels each column: counting number, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first column labels each row: multiples of 2, multiples of 3, multiples of 4, multiples of 5, multiples of 6, multiples of 7, multiples of 8, multiples of 9. The column labeled 1 has the following values: 2, 3, 4, 5, 6, 7, 8, 9. The column labeled 2 has the following values: 4, 6, 8, 10, 12, 14, 16, 18. The column labeled 3 has the following values: 6, 9, 12, 15, 18, 21, 24, 27. The column labeled 4 has the following values: 8, 12, 16, 20, 24, 28, 32, 36. The column labeled 5 has the following values: 10, 15, 20, 25, 30, 35, 40, 45. The column labeled 6 has the following values: 12, 18, 24, 30, 36, 42, 48, 54. The column labeled 7 has the following values: 14, 21, 28, 35, 42, 49, 56, 63. The column labeled 8 has the following values: 16, 24, 32, 40, 48, 56, 64, 72. The column labeled 9 has the following values: 18, 27, 36, 45, 54, 63, 72, 81. The column labeled 10 has the following values: 20, 30, 40, 50, 60, 70, 80, 90. The column labeled 11 has the following values: 22, 33, 44, 55, 66, 77, 88, 99. The column labeled 12 has the following values: 24, 36, 48, 60, 72, 84, 96, 108.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"left\">Counting Number<\/th>\n<th data-valign=\"middle\" data-align=\"left\">1<\/th>\n<th data-valign=\"middle\" data-align=\"left\">2<\/th>\n<th data-valign=\"middle\" data-align=\"left\">3<\/th>\n<th data-valign=\"middle\" data-align=\"left\">4<\/th>\n<th data-valign=\"middle\" data-align=\"left\">5<\/th>\n<th data-valign=\"middle\" data-align=\"left\">6<\/th>\n<th data-valign=\"middle\" data-align=\"left\">7<\/th>\n<th data-valign=\"middle\" data-align=\"left\">8<\/th>\n<th data-valign=\"middle\" data-align=\"left\">9<\/th>\n<th data-valign=\"middle\" data-align=\"left\">10<\/th>\n<th data-valign=\"middle\" data-align=\"left\">11<\/th>\n<th data-valign=\"middle\" data-align=\"left\">12<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 2<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">2<\/td>\n<td data-valign=\"middle\" data-align=\"left\">4<\/td>\n<td data-valign=\"middle\" data-align=\"left\">6<\/td>\n<td data-valign=\"middle\" data-align=\"left\">8<\/td>\n<td data-valign=\"middle\" data-align=\"left\">10<\/td>\n<td data-valign=\"middle\" data-align=\"left\">12<\/td>\n<td data-valign=\"middle\" data-align=\"left\">14<\/td>\n<td data-valign=\"middle\" data-align=\"left\">16<\/td>\n<td data-valign=\"middle\" data-align=\"left\">18<\/td>\n<td data-valign=\"middle\" data-align=\"left\">20<\/td>\n<td data-valign=\"middle\" data-align=\"left\">22<\/td>\n<td data-valign=\"middle\" data-align=\"left\">24<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 3<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">3<\/td>\n<td data-valign=\"middle\" data-align=\"left\">6<\/td>\n<td data-valign=\"middle\" data-align=\"left\">9<\/td>\n<td data-valign=\"middle\" data-align=\"left\">12<\/td>\n<td data-valign=\"middle\" data-align=\"left\">15<\/td>\n<td data-valign=\"middle\" data-align=\"left\">18<\/td>\n<td data-valign=\"middle\" data-align=\"left\">21<\/td>\n<td data-valign=\"middle\" data-align=\"left\">24<\/td>\n<td data-valign=\"middle\" data-align=\"left\">27<\/td>\n<td data-valign=\"middle\" data-align=\"left\">30<\/td>\n<td data-valign=\"middle\" data-align=\"left\">33<\/td>\n<td data-valign=\"middle\" data-align=\"left\">36<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 4<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">4<\/td>\n<td data-valign=\"middle\" data-align=\"left\">8<\/td>\n<td data-valign=\"middle\" data-align=\"left\">12<\/td>\n<td data-valign=\"middle\" data-align=\"left\">16<\/td>\n<td data-valign=\"middle\" data-align=\"left\">20<\/td>\n<td data-valign=\"middle\" data-align=\"left\">24<\/td>\n<td data-valign=\"middle\" data-align=\"left\">28<\/td>\n<td data-valign=\"middle\" data-align=\"left\">32<\/td>\n<td data-valign=\"middle\" data-align=\"left\">36<\/td>\n<td data-valign=\"middle\" data-align=\"left\">40<\/td>\n<td data-valign=\"middle\" data-align=\"left\">44<\/td>\n<td data-valign=\"middle\" data-align=\"left\">48<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 5<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">5<\/td>\n<td data-valign=\"middle\" data-align=\"left\">10<\/td>\n<td data-valign=\"middle\" data-align=\"left\">15<\/td>\n<td data-valign=\"middle\" data-align=\"left\">20<\/td>\n<td data-valign=\"middle\" data-align=\"left\">25<\/td>\n<td data-valign=\"middle\" data-align=\"left\">30<\/td>\n<td data-valign=\"middle\" data-align=\"left\">35<\/td>\n<td data-valign=\"middle\" data-align=\"left\">40<\/td>\n<td data-valign=\"middle\" data-align=\"left\">45<\/td>\n<td data-valign=\"middle\" data-align=\"left\">50<\/td>\n<td data-valign=\"middle\" data-align=\"left\">55<\/td>\n<td data-valign=\"middle\" data-align=\"left\">60<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 6<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">6<\/td>\n<td data-valign=\"middle\" data-align=\"left\">12<\/td>\n<td data-valign=\"middle\" data-align=\"left\">18<\/td>\n<td data-valign=\"middle\" data-align=\"left\">24<\/td>\n<td data-valign=\"middle\" data-align=\"left\">30<\/td>\n<td data-valign=\"middle\" data-align=\"left\">36<\/td>\n<td data-valign=\"middle\" data-align=\"left\">42<\/td>\n<td data-valign=\"middle\" data-align=\"left\">48<\/td>\n<td data-valign=\"middle\" data-align=\"left\">54<\/td>\n<td data-valign=\"middle\" data-align=\"left\">60<\/td>\n<td data-valign=\"middle\" data-align=\"left\">66<\/td>\n<td data-valign=\"middle\" data-align=\"left\">72<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 7<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">7<\/td>\n<td data-valign=\"middle\" data-align=\"left\">14<\/td>\n<td data-valign=\"middle\" data-align=\"left\">21<\/td>\n<td data-valign=\"middle\" data-align=\"left\">28<\/td>\n<td data-valign=\"middle\" data-align=\"left\">35<\/td>\n<td data-valign=\"middle\" data-align=\"left\">42<\/td>\n<td data-valign=\"middle\" data-align=\"left\">49<\/td>\n<td data-valign=\"middle\" data-align=\"left\">56<\/td>\n<td data-valign=\"middle\" data-align=\"left\">63<\/td>\n<td data-valign=\"middle\" data-align=\"left\">70<\/td>\n<td data-valign=\"middle\" data-align=\"left\">77<\/td>\n<td data-valign=\"middle\" data-align=\"left\">84<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 8<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">8<\/td>\n<td data-valign=\"middle\" data-align=\"left\">16<\/td>\n<td data-valign=\"middle\" data-align=\"left\">24<\/td>\n<td data-valign=\"middle\" data-align=\"left\">32<\/td>\n<td data-valign=\"middle\" data-align=\"left\">40<\/td>\n<td data-valign=\"middle\" data-align=\"left\">48<\/td>\n<td data-valign=\"middle\" data-align=\"left\">56<\/td>\n<td data-valign=\"middle\" data-align=\"left\">64<\/td>\n<td data-valign=\"middle\" data-align=\"left\">72<\/td>\n<td data-valign=\"middle\" data-align=\"left\">80<\/td>\n<td data-valign=\"middle\" data-align=\"left\">88<\/td>\n<td data-valign=\"middle\" data-align=\"left\">96<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Multiples of 9<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">9<\/td>\n<td data-valign=\"middle\" data-align=\"left\">18<\/td>\n<td data-valign=\"middle\" data-align=\"left\">27<\/td>\n<td data-valign=\"middle\" data-align=\"left\">36<\/td>\n<td data-valign=\"middle\" data-align=\"left\">45<\/td>\n<td data-valign=\"middle\" data-align=\"left\">54<\/td>\n<td data-valign=\"middle\" data-align=\"left\">63<\/td>\n<td data-valign=\"middle\" data-align=\"left\">72<\/td>\n<td data-valign=\"middle\" data-align=\"left\">81<\/td>\n<td data-valign=\"middle\" data-align=\"left\">90<\/td>\n<td data-valign=\"middle\" data-align=\"left\">99<\/td>\n<td data-valign=\"middle\" data-align=\"left\">108<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"note\" id=\"fs-id1167836548764\">\n<div data-type=\"title\">Multiple of a Number<\/div>\n<p id=\"fs-id1167836507946\">A number is a <strong data-effect=\"bold\">multiple<\/strong> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> if it is the product of a counting number and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ecab3f5df4767e23a2660ceb88ceabd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167836700519\">Another way to say that 15 is a multiple of 3 is to say that 15 is <span data-type=\"term\">divisible<\/span> by 3. That means that when we divide 3 into 15, we get a counting number. In fact, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed5bd88c47b44efb688f16a96ce3431a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&divide;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"26\" style=\"vertical-align: -1px;\" \/> is 5, so 15 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-996ce8b1b57257395d76490510975b7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&middot;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167836362413\">\n<div data-type=\"title\">Divisible by a Number<\/div>\n<p id=\"fs-id1167836447704\">If a number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> is a multiple of <em data-effect=\"italics\">n<\/em>, then <em data-effect=\"italics\">m<\/em> is <strong data-effect=\"bold\">divisible<\/strong> by <em data-effect=\"italics\">n<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1167836415616\">If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:<\/p>\n<div data-type=\"note\" id=\"fs-id1167836525210\">\n<div data-type=\"title\">Divisibility Tests<\/div>\n<p id=\"fs-id1167829693509\">A number is divisible by:<\/p>\n<p id=\"fs-id1167836447272\">\u2003\u2003\u20032 if the last digit is 0, 2, 4, 6, or 8.<\/p>\n<p id=\"fs-id1167836486860\">\u2003\u2003\u20033 if the sum of the digits is divisible by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a650b6362973a0817ee43187a6682e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167836534606\">\u2003\u2003\u20035 if the last digit is 5 or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d077f1202cae43a1855e4e1bb5939948_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167836510672\">\u2003\u2003\u20036 if it is divisible by both 2 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a650b6362973a0817ee43187a6682e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167836608078\">\u2003\u2003\u200310 if it ends with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d077f1202cae43a1855e4e1bb5939948_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167833047554\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836516080\">\n<div data-type=\"problem\" id=\"fs-id1167836558662\">\n<p id=\"fs-id1167833053866\">Is 5,625 divisible by <span class=\"token\">\u24d0<\/span> 2? <span class=\"token\">\u24d1<\/span> 3? <span class=\"token\">\u24d2<\/span> 5 or 10? <span class=\"token\">\u24d3<\/span> 6?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836579398\">\n<ol id=\"fs-id1167836510387\" type=\"1\" class=\"circled\">\n<li><span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd8518ef3baa4df4d78f8f9ca6fab3d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#53;&#44;&#54;&#50;&#53;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#50;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#101;&#115;&#32;&#105;&#116;&#32;&#101;&#110;&#100;&#32;&#105;&#110;&#32;&#48;&#44;&#32;&#50;&#44;&#32;&#52;&#44;&#32;&#54;&#32;&#111;&#114;&#32;&#56;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#44;&#54;&#50;&#53;&#32;&#105;&#115;&#32;&#110;&#111;&#116;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#50;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"104\" width=\"551\" style=\"vertical-align: -47px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/li>\n<li><span class=\"token\">\u24d1<\/span>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f74742d5858eac58cc3b6b5a153aca5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#53;&#44;&#54;&#50;&#53;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#51;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#97;&#116;&#32;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#115;&#117;&#109;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#103;&#105;&#116;&#115;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#43;&#54;&#43;&#50;&#43;&#53;&#61;&#49;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#32;&#115;&#117;&#109;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#51;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#89;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#44;&#54;&#50;&#53;&#32;&#105;&#115;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#51;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"126\" width=\"528\" style=\"vertical-align: -58px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/li>\n<li><span class=\"token\">\u24d2<\/span>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e117352c83951abf836c3978df278ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#53;&#44;&#54;&#50;&#53;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#53;&#32;&#111;&#114;&#32;&#49;&#48;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#97;&#116;&#32;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#115;&#116;&#32;&#100;&#105;&#103;&#105;&#116;&#63;&#32;&#73;&#116;&#32;&#105;&#115;&#32;&#53;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#44;&#54;&#50;&#53;&#32;&#105;&#115;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#53;&#32;&#98;&#117;&#116;&#32;&#110;&#111;&#116;&#32;&#98;&#121;&#32;&#49;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"637\" style=\"vertical-align: -36px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/li>\n<li><span class=\"token\">\u24d3<\/span>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b42d7f3a836bdb6a479a09c39a0ef96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#53;&#44;&#54;&#50;&#53;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#54;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#105;&#116;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#98;&#111;&#116;&#104;&#32;&#50;&#32;&#97;&#110;&#100;&#32;&#51;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#44;&#32;&#53;&#44;&#54;&#50;&#53;&#32;&#105;&#115;&#32;&#110;&#111;&#116;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#50;&#44;&#32;&#115;&#111;&#32;&#53;&#44;&#54;&#50;&#53;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#116;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#98;&#108;&#101;&#32;&#98;&#121;&#32;&#54;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"104\" width=\"682\" style=\"vertical-align: -47px;\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836691875\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829691614\">\n<div data-type=\"problem\" id=\"fs-id1167836356400\">\n<p id=\"fs-id1167829694806\">Is 4,962 divisible by <span class=\"token\">\u24d0<\/span> 2? <span class=\"token\">\u24d1<\/span> 3? <span class=\"token\">\u24d2<\/span> 5? <span class=\"token\">\u24d3<\/span> 6? <span class=\"token\">\u24d4<\/span> 10?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836342477\">\n<p id=\"fs-id1167829694818\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> yes<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span> no<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833019387\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836311134\">\n<div data-type=\"problem\" id=\"fs-id1167836551933\">\n<p id=\"fs-id1167836293370\">Is 3,765 divisible by <span class=\"token\">\u24d0<\/span> 2? <span class=\"token\">\u24d1<\/span> 3? <span class=\"token\">\u24d2<\/span> 5? <span class=\"token\">\u24d3<\/span> 6? <span class=\"token\">\u24d4<\/span> 10?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836341297\">\n<p id=\"fs-id1167836559793\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> yes <span class=\"token\">\u24d3<\/span> no<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span> no<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836511506\">In mathematics, there are often several ways to talk about the same ideas. So far, we\u2019ve seen that if <em data-effect=\"italics\">m<\/em> is a multiple of <em data-effect=\"italics\">n<\/em>, we can say that <em data-effect=\"italics\">m<\/em> is divisible by <em data-effect=\"italics\">n<\/em>. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.<\/p>\n<p id=\"fs-id1167836612806\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d4df87e1ca38a9da3b3f4202781339d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&middot;&#57;&#61;&#55;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\" \/> we say that 8 and 9 are <span data-type=\"term\">factors<\/span> of 72. When we write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39b53f7d61c5306bbd030f11c30e20b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#50;&#61;&#56;&middot;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\" \/> we say we have factored 72.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833056551\" data-alt=\"8 times 9 is 72. 8 and 9 are factors. 72 is the product.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"8 times 9 is 72. 8 and 9 are factors. 72 is the product.\" \/><\/span><\/p>\n<p id=\"fs-id1167836387255\">Other ways to factor 72 are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c3c144220af0c461c06a3deecd5a9e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&middot;&#55;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&middot;&#51;&#54;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&middot;&#50;&#52;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&middot;&#49;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"176\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b100744124a1936aa7a8b4125a225fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&middot;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> The number 72 has many factors: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a62b6de30f01ea87b8f0007ea9be4fe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#54;&#44;&#56;&#44;&#57;&#44;&#49;&#50;&#44;&#49;&#56;&#44;&#50;&#52;&#44;&#51;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"215\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dfc2b7e9007d9789664ffb483c35d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167836539545\">\n<div data-type=\"title\">Factors<\/div>\n<p id=\"fs-id1167836322020\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4752e3c7f6f79fe949cc961f5a008188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are <strong data-effect=\"bold\">factors<\/strong> of <em data-effect=\"italics\">m<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1167836387232\">Some numbers, such as 72, have many factors. Other numbers have only two factors. A <span data-type=\"term\">prime number<\/span> is a counting number greater than 1 whose only factors are 1 and itself.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836410498\">\n<div data-type=\"title\">Prime number and Composite number<\/div>\n<p id=\"fs-id1167836538134\">A <strong data-effect=\"bold\">prime number<\/strong> is a counting number greater than 1 whose only factors are 1 and the number itself.<\/p>\n<p id=\"fs-id1167829695314\">A <strong data-effect=\"bold\">composite number<\/strong> is a counting number that is not prime. A composite number has factors other than 1 and the number itself.<\/p>\n<\/div>\n<p id=\"fs-id1167836515162\">The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the \u201cprime\u201d or \u201ccomposite\u201d label for each!<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836418575\" data-alt=\"This table has three columns, 19 rows and a header row. The header row labels each column: number, factors and prime or composite. The values in each row are as follows: number 2, factors 1, 2, prime; number 3, factors 1, 3, prime; number 4, factors 1, 2, 4, composite; number 5, factors, 1, 5, prime; number 6, factors 1, 2, 3, 6, composite; number 7, factors 1, 7, prime; number 8, factors 1, 2, 4, 8, composite; number 9, factors 1, 3, 9, composite; number 10, factors 1, 2, 5, 10, composite; number 11, factors 1, 11, prime; number 12, factors 1, 2, 3, 4, 6, 12, composite; number 13, factors 1, 13, prime; number 14, factors 1, 2, 7, 14, composite; number 15, factors 1, 3, 5, 15, composite; number 16, factors 1, 2, 4, 8, 16, composite; number 17, factors 1, 17, prime; number 18, factors 1, 2, 3, 6, 9, 18, composite; number 19, factors 1, 19, prime; number 20, factors 1, 2, 4, 5, 10, 20, composite.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_004_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has three columns, 19 rows and a header row. The header row labels each column: number, factors and prime or composite. The values in each row are as follows: number 2, factors 1, 2, prime; number 3, factors 1, 3, prime; number 4, factors 1, 2, 4, composite; number 5, factors, 1, 5, prime; number 6, factors 1, 2, 3, 6, composite; number 7, factors 1, 7, prime; number 8, factors 1, 2, 4, 8, composite; number 9, factors 1, 3, 9, composite; number 10, factors 1, 2, 5, 10, composite; number 11, factors 1, 11, prime; number 12, factors 1, 2, 3, 4, 6, 12, composite; number 13, factors 1, 13, prime; number 14, factors 1, 2, 7, 14, composite; number 15, factors 1, 3, 5, 15, composite; number 16, factors 1, 2, 4, 8, 16, composite; number 17, factors 1, 17, prime; number 18, factors 1, 2, 3, 6, 9, 18, composite; number 19, factors 1, 19, prime; number 20, factors 1, 2, 4, 5, 10, 20, composite.\" \/><\/span><\/p>\n<p id=\"fs-id1167829694566\">The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.<\/p>\n<p id=\"fs-id1167833060885\">A composite number can be written as a unique product of primes. This is called the <span data-type=\"term\">prime factorization<\/span> of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836530500\">\n<div data-type=\"title\">Prime Factorization<\/div>\n<p id=\"fs-id1167836628445\">The <strong data-effect=\"bold\">prime factorization<\/strong> of a number is the product of prime numbers that equals the number.<\/p>\n<\/div>\n<p id=\"fs-id1167836546976\">To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.<\/p>\n<p id=\"fs-id1167836334529\">If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829937177\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Find the Prime Factorization of a Composite Number<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829599878\">\n<div data-type=\"problem\" id=\"fs-id1167829599880\">\n<p id=\"fs-id1167829599883\">Factor 48.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833056756\"><span data-type=\"media\" id=\"fs-id1167833056758\" data-alt=\"Step 1 is to find two factors whose product is 48 and use these numbers to create two branches. The two branches originating from 48 are formed by the factors 2 and 24.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find two factors whose product is 48 and use these numbers to create two branches. The two branches originating from 48 are formed by the factors 2 and 24.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836542295\" data-alt=\"Step 2 is to circle the prime factor. This completes that branch. In this case, 2 is circled as it is prime.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to circle the prime factor. This completes that branch. In this case, 2 is circled as it is prime.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836513367\" data-alt=\"Step 3 is to treat the composite factor as a product, break it into two more factors and continue the process. 24 is not prime. It is broken into 4 and 6. 4 and 6 are not prime. 4 is broken into its factors 2 and 2, both of which are circled. 6 is not prime. It is broken into factors 2 and 3, both of which are circled.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to treat the composite factor as a product, break it into two more factors and continue the process. 24 is not prime. It is broken into 4 and 6. 4 and 6 are not prime. 4 is broken into its factors 2 and 2, both of which are circled. 6 is not prime. It is broken into factors 2 and 3, both of which are circled.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836507776\" data-alt=\"Step 4 is to write the original composite number as the product of all the circled primes. 48 is 2 into 2 into 2 into 2 into 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_005d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to write the original composite number as the product of all the circled primes. 48 is 2 into 2 into 2 into 2 into 3.\" \/><\/span><\/p>\n<p id=\"fs-id1171790680445\">\n<div data-type=\"newline\"><\/div>\n<p id=\"fs-id1167836545840\">We say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-768495cc22379539dc63613224ccb4cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"45\" style=\"vertical-align: 0px;\" \/> is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.<\/p>\n<p id=\"fs-id1167836516314\">If we first factored 48 in a different way, for example as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f7829372d4e6f4cd74b2f6967585b7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&middot;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"22\" style=\"vertical-align: -4px;\" \/> the result would still be the same. Finish the prime factorization and verify this for yourself.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836506869\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836506872\">\n<div data-type=\"problem\" id=\"fs-id1167836558433\">\n<p id=\"fs-id1167836558436\">Find the prime factorization of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3c6c0d94f2823edf0ccf50f18a0c78b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829694799\">\n<p id=\"fs-id1167829694801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4498aff1421f7f235dedd1ff3eeb6488_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836546878\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836418325\">\n<div data-type=\"problem\" id=\"fs-id1167836418327\">\n<p id=\"fs-id1167836418329\">Find the prime factorization of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-626b65963604e0a63e1cb13349aa7181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836418464\">\n<p id=\"fs-id1167836418466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be549a5f038fd2e88e4d5ddfb1c6b616_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#51;&middot;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836386665\" class=\"howto\">\n<div data-type=\"title\">Find the prime factorization of a composite number.<\/div>\n<ol id=\"fs-id1167836610439\" type=\"1\" class=\"stepwise\">\n<li>Find two factors whose product is the given number, and use these numbers to create two branches.<\/li>\n<li>If a factor is prime, that branch is complete. Circle the prime, like a leaf on the tree.<\/li>\n<li>If a factor is not prime, write it as the product of two factors and continue the process.<\/li>\n<li>Write the composite number as the product of all the circled primes.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836693029\">One of the reasons we look at primes is to use these techniques to find the <span data-type=\"term\">least common multiple<\/span> of two numbers. This will be useful when we add and subtract fractions with different denominators.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836287670\">\n<div data-type=\"title\">Least Common Multiple<\/div>\n<p id=\"fs-id1167833009881\">The <strong data-effect=\"bold\">least common multiple (LCM)<\/strong> of two numbers is the smallest number that is a multiple of both numbers.<\/p>\n<\/div>\n<p id=\"fs-id1167836558236\">To find the least common multiple of two numbers we will use the Prime Factors Method. Let\u2019s find the LCM of 12 and 18 using their prime factors.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829937221\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Find the Least Common Multiple Using the Prime Factors Method<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836506713\">\n<div data-type=\"problem\" id=\"fs-id1167836506715\">\n<p id=\"fs-id1167829691676\">Find the least common multiple (LCM) of 12 and 18 using the prime factors method.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829691681\"><span data-type=\"media\" id=\"fs-id1167836610203\" data-alt=\"Step 1 is to write each number as a product of primes. The number 12 is written as a product of 2, 2 and 3. The number 18 is written as a product of 2, 3 and 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write each number as a product of primes. The number 12 is written as a product of 2, 2 and 3. The number 18 is written as a product of 2, 3 and 3.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829692860\" data-alt=\"Step 2 is to list the primes of each number such that primes are vertically matched when possible. The factors of 12 are listed as 2, 2 and 3. The factors of 18 are written below this. The first 2 at the top lines up with the first two at the bottom. The second 2 at the top does not line up with anything. The 3 at the top lines up with a 3 at the bottom. The last 3 at the bottom does not line up with anything. Hence, four columns are made.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to list the primes of each number such that primes are vertically matched when possible. The factors of 12 are listed as 2, 2 and 3. The factors of 18 are written below this. The first 2 at the top lines up with the first two at the bottom. The second 2 at the top does not line up with anything. The 3 at the top lines up with a 3 at the bottom. The last 3 at the bottom does not line up with anything. Hence, four columns are made.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836622827\" data-alt=\"Step 3 is to bring down the number from each column. When a column has the same number at the top and the bottom, that number is brought down. When a column has only one number that number is brought down. The numbers brought down are 2, 2, 3 and 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to bring down the number from each column. When a column has the same number at the top and the bottom, that number is brought down. When a column has only one number that number is brought down. The numbers brought down are 2, 2, 3 and 3.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836608172\" data-alt=\"Step 4 is to multiply the factors. The numbers brought down are multiplied with each other to get the LCM. The LCM is 2 into 2 into 3 into 3 equal to 36.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_006d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to multiply the factors. The numbers brought down are multiplied with each other to get the LCM. The LCM is 2 into 2 into 3 into 3 equal to 36.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836387125\">Notice that the prime factors of 12 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e5677dbd7d8c44e28ac4f5d946fc1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&middot;&#50;&middot;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> and the prime factors of 18 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da0b99c69678420a7c7cd56e34caa50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&middot;&#51;&middot;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> are included in the LCM <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4ad913924efc5732f7e237bc9fe851f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&middot;&#50;&middot;&#51;&middot;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> So 36 is the least common multiple of 12 and 18.<\/p>\n<p id=\"fs-id1167829691476\">By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the <em data-effect=\"italics\">least<\/em> common multiple.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836693644\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836693647\">\n<div data-type=\"problem\" id=\"fs-id1167836693649\">\n<p id=\"fs-id1167836556107\">Find the LCM of 9 and 12 using the Prime Factors Method.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836556111\">\n<p id=\"fs-id1167836447731\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dce5a8d8a80b6aa9c2a9f6810851e582_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836556037\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836706707\">\n<div data-type=\"problem\" id=\"fs-id1167836706710\">\n<p id=\"fs-id1167836706712\">Find the LCM of 18 and 24 using the Prime Factors Method.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833279842\">\n<p id=\"fs-id1167833279844\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7bef200c216bbe8be23cd2a2c1371d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829597058\" class=\"howto\">\n<div data-type=\"title\">Find the least common multiple using the Prime Factors Method.<\/div>\n<ol id=\"fs-id1167836317446\" type=\"1\" class=\"stepwise\">\n<li>Write each number as a product of primes.<\/li>\n<li>List the primes of each number. Match primes vertically when possible.<\/li>\n<li>Bring down the columns.<\/li>\n<li>Multiply the factors.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836514368\">\n<h3 data-type=\"title\">Use Variables and Algebraic Symbols<\/h3>\n<p id=\"fs-id1167836550860\">In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a <span data-type=\"term\">variable<\/span> and letters commonly used for variables are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df210970d1d9c1c2214ba1f3053da7f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#121;&#44;&#97;&#44;&#98;&#44;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167829586674\">\n<div data-type=\"title\">Variable<\/div>\n<p id=\"fs-id1167836558388\">A <strong data-effect=\"bold\">variable<\/strong> is a letter that represents a number whose value may change.<\/p>\n<\/div>\n<p id=\"fs-id1167829694184\">A number whose value always remains the same is called a <span data-type=\"term\">constant<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836597954\">\n<div data-type=\"title\">Constant<\/div>\n<p id=\"fs-id1167833023164\">A <strong data-effect=\"bold\">constant<\/strong> is a number whose value always stays the same.<\/p>\n<\/div>\n<p id=\"fs-id1167833383070\">To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We\u2019ll list the symbols used to indicate these operations below.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833383073\">\n<div data-type=\"title\">Operation Symbols<\/div>\n<table id=\"fs-id1167836551950\" class=\"unnumbered\" summary=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: operation, notation, say and the result is. Row 1 has the following entries: addition, a plus b, a plus b and the sum of a and b. Row 2 has the following entries: subtraction, a minus b, a minus b and the difference of a and b. Row 3 has the following entries: multiplication, notations a dot b, ab, open parentheses a close parentheses open parentheses b close parentheses, open parentheses a close parentheses b, a open parentheses b close parentheses. say a times b, the product of a and b. Row 4 has the following entries: division, a divided by b, a slash b, a upon b and b right parentheses a overbar, say a divided by b, the quotient of a and b; a is called the dividend, and b is called the divisor.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Operation<\/th>\n<th data-valign=\"top\" data-align=\"left\">Notation<\/th>\n<th data-valign=\"top\" data-align=\"left\">Say:<\/th>\n<th data-valign=\"top\" data-align=\"left\">The result is\u2026<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Addition<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38830949a60ce6786a6fdf6309482002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> plus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtraction<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b644d18119c5701b308f883bd50656d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> minus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiplication<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6848c2359b48af3f5f34fecf40870b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;&#44;&#97;&#98;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7aa6efad3948cefc6187e6af0324c06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#98;&#44;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Division<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01eac8e271d3f70e36776ff9e70f32e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&divide;&#98;&#44;&#97;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#98;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"106\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">the quotient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-deb416b5663cc095ad296e8ae84bfa4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"12\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is called the dividend, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is called the divisor<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"fs-id1167836558375\">When two quantities have the same value, we say they are equal and connect them with an <span data-type=\"term\" class=\"no-emphasis\">equal<\/span> sign.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836614086\">\n<div data-type=\"title\">Equality Symbol<\/div>\n<p id=\"fs-id1167836614092\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e69295ccf8d8ca292abcc97f861e345f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/> is read \u201c<em data-effect=\"italics\">a<\/em> is equal to <em data-effect=\"italics\">b<\/em>.\u201d<\/p>\n<p id=\"fs-id1167836531940\">The symbol \u201c=\u201d is called the equal sign.<\/p>\n<\/div>\n<p id=\"fs-id1167836531945\">On the <span data-type=\"term\" class=\"no-emphasis\">number line<\/span>, the numbers get larger as they go from left to right. The number line can be used to explain the symbols \u201c&lt;\u201d and \u201c&gt;\u201d.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833056435\">\n<div data-type=\"title\">Inequality<\/div>\n<p><span data-type=\"media\" id=\"fs-id1171791260555\" data-alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_019_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\" \/><\/span><\/div>\n<p id=\"fs-id1167833397294\">The expressions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b57677a3c7d31d00aea3cdd09443b09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b84b1480aef6484626cffaeccec0b9fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/> can be read from left to right or right to left, though in English we usually read from left to right. In general,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836624800\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4550902a1d3e6adbc2a19a6bff0f179b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#60;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#62;&#97;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#111;&#114;&#32;&#101;&#120;&#97;&#109;&#112;&#108;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&#60;&#49;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#49;&#62;&#55;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#62;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#60;&#97;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#111;&#114;&#32;&#101;&#120;&#97;&#109;&#112;&#108;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#62;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#60;&#49;&#55;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"562\" style=\"vertical-align: -14px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1167836618823\">\n<div data-type=\"title\">Inequality Symbols<\/div>\n<table id=\"fs-id1167836510455\" class=\"unnumbered\" summary=\"The table describes inequality symbols in words. The symbols described are a is not equal to b, a is less than b, a is less than or equal to b, a is greater then b, a is greater than or equal to b.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Inequality Symbols<\/th>\n<th data-valign=\"top\" data-align=\"left\">Words<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8740efbef2c653ea7d353859ecc91afd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">not equal to b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b57677a3c7d31d00aea3cdd09443b09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22bb7f496e0a4516393886afb53ffa7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than or equal to b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b84b1480aef6484626cffaeccec0b9fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4925aae9d8237338b0824392ba2d182a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#103;&#101;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than or equal to b.<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"fs-id1167836447550\">Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an <span data-type=\"term\">expression<\/span>, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829597232\">\n<div data-type=\"title\">Grouping Symbols<\/div>\n<div data-type=\"equation\" id=\"fs-id1171790895306\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4909a61294f5368dc2c706f498146843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#114;&#101;&#110;&#116;&#104;&#101;&#115;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#107;&#101;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"193\" style=\"vertical-align: -27px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836521188\">Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836521192\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-586531a01ef6eb8d16212e82bad10bc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#49;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#91;&#50;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&divide;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#49;&#51;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"608\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"fs-id1167833412639\">What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have <em data-effect=\"italics\">expressions<\/em> and <em data-effect=\"italics\">equations<\/em>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833412655\">\n<div data-type=\"title\">Expression<\/div>\n<p id=\"fs-id1167836530664\">An <strong data-effect=\"bold\">expression<\/strong> is a number, a variable, or a combination of numbers and variables using operation symbols.<\/p>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1167826987885\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3765b01e96f99a27667d70b33eeab6c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#120;&#112;&#114;&#101;&#115;&#115;&#105;&#111;&#110;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#111;&#114;&#100;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#110;&#103;&#108;&#105;&#115;&#104;&#32;&#80;&#104;&#114;&#97;&#115;&#101;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#112;&#108;&#117;&#115;&#32;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#115;&#117;&#109;&#32;&#111;&#102;&#32;&#116;&#104;&#114;&#101;&#101;&#32;&#97;&#110;&#100;&#32;&#102;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#110;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#117;&#115;&#32;&#111;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#111;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&middot;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#115;&#105;&#120;&#32;&#97;&#110;&#100;&#32;&#115;&#101;&#118;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#118;&#105;&#100;&#101;&#100;&#32;&#98;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"110\" width=\"574\" style=\"vertical-align: -52px;\" \/><\/div>\n<p id=\"fs-id1167836524937\">Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.<\/p>\n<p id=\"fs-id1167836524941\">An <span data-type=\"term\">equation<\/span> is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836510026\">\n<div data-type=\"title\">Equation<\/div>\n<p id=\"fs-id1167836510031\">An <strong data-effect=\"bold\">equation<\/strong> is two expressions connected by an equal sign.<\/p>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1167835303120\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a5f5ad0f60da50aa5dcc2333790cc0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#110;&#103;&#108;&#105;&#115;&#104;&#32;&#83;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#43;&#53;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#115;&#117;&#109;&#32;&#111;&#102;&#32;&#116;&#104;&#114;&#101;&#101;&#32;&#97;&#110;&#100;&#32;&#102;&#105;&#118;&#101;&#32;&#105;&#115;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#101;&#105;&#103;&#104;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#110;&#45;&#49;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#117;&#115;&#32;&#111;&#110;&#101;&#32;&#101;&#113;&#117;&#97;&#108;&#115;&#32;&#102;&#111;&#117;&#114;&#116;&#101;&#101;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&middot;&#55;&#61;&#52;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#115;&#105;&#120;&#32;&#97;&#110;&#100;&#32;&#115;&#101;&#118;&#101;&#110;&#32;&#105;&#115;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#102;&#111;&#114;&#116;&#121;&#45;&#116;&#119;&#111;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#53;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#102;&#105;&#102;&#116;&#121;&#45;&#116;&#104;&#114;&#101;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#43;&#57;&#61;&#50;&#121;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#121;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#108;&#117;&#115;&#32;&#110;&#105;&#110;&#101;&#32;&#105;&#115;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#116;&#119;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#117;&#115;&#32;&#116;&#104;&#114;&#101;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"127\" width=\"586\" style=\"vertical-align: -59px;\" \/><\/div>\n<p id=\"fs-id1167836493942\">Suppose we need to multiply 2 nine times. We could write this as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-004f7b4aee5a639d82920633982dcbde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"85\" style=\"vertical-align: 0px;\" \/> This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2996da55f7dbeabb41b65160a0e5d9ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a402f8a62a07dd1eb366927c0a3e64f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acdd4dc3cf725cd78d89c4e77b97e3f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"80\" style=\"vertical-align: 0px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93dc4b3298d2eeb792cb14a1628abad6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: 0px;\" \/> In expressions such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-538e13f145160df1011676bf5f93dcc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"21\" style=\"vertical-align: -4px;\" \/> the 2 is called the <em data-effect=\"italics\">base<\/em> and the 3 is called the <em data-effect=\"italics\">exponent<\/em>. The <span data-type=\"term\" class=\"no-emphasis\">exponent<\/span> tells us how many times we need to multiply the <span data-type=\"term\" class=\"no-emphasis\">base<\/span>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836321231\" data-alt=\"The expression shows the number 2, with the number 3 written to its top right. 2 is labeled base and 3 is labeled exponent. This means multiply 2 by itself, three times, as in 2 times 2 times 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_007_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression shows the number 2, with the number 3 written to its top right. 2 is labeled base and 3 is labeled exponent. This means multiply 2 by itself, three times, as in 2 times 2 times 2.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1167833059322\">\n<div data-type=\"title\">Exponential Notation<\/div>\n<p id=\"fs-id1167833059328\">We say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a402f8a62a07dd1eb366927c0a3e64f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> is in <em data-effect=\"italics\">exponential notation<\/em> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2996da55f7dbeabb41b65160a0e5d9ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> is in <em data-effect=\"italics\">expanded notation<\/em>.<\/p>\n<p id=\"fs-id1167836286079\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef78644f25c125ead81d3c69ef32dee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> means multiply <em data-effect=\"italics\">a<\/em> by itself, <em data-effect=\"italics\">n<\/em> times.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833059485\" data-alt=\"The expression shown is a to the nth power. Here a is the base and n is the exponent. This is equal to a times a times a and so on, repeated n times. This has n factors.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_008_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression shown is a to the nth power. Here a is the base and n is the exponent. This is equal to a times a times a and so on, repeated n times. This has n factors.\" \/><\/span><\/p>\n<p id=\"fs-id1167836329008\">The expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef78644f25c125ead81d3c69ef32dee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> is read <em data-effect=\"italics\">a<\/em> to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02553ae49d720703c253c7bcbf008617_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#116;&#104;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"24\" style=\"vertical-align: 0px;\" \/> power.<\/p>\n<\/div>\n<p id=\"fs-id1167836509491\">While we read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef78644f25c125ead81d3c69ef32dee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8133261f86cd6f70b81d192d575ccd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#96;&#96;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02553ae49d720703c253c7bcbf008617_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#116;&#104;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"24\" style=\"vertical-align: 0px;\" \/> power\u201d, we usually read:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829597038\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dcefc1b68de1d6ebb07703bb59628c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#96;&#96;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#113;&#117;&#97;&#114;&#101;&#100;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#96;&#96;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#117;&#98;&#101;&#100;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"187\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"fs-id1167836493073\">We\u2019ll see later why <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91fa2bda4c48f12d3beccdcacaba4770_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9c49c3f57f6f5eed5f4192bf868fe18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> have special names.<\/p>\n<p id=\"fs-id1167836493093\"><a href=\"#fs-id1167833021966\" class=\"autogenerated-content\">(Figure)<\/a> shows how we read some expressions with exponents.<\/p>\n<table id=\"fs-id1167833021966\" summary=\"This table shows four expressions and words to describe these. The expressions described are 7 to the second power or 7 squared, 5 to the third power or 5 cubed, 9 to the fourth power and 12 to the fifth.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Expression<\/th>\n<th data-valign=\"top\" data-align=\"left\">In Words<\/th>\n<th data-valign=\"top\" data-align=\"left\"><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">7<sup>2<\/sup><\/td>\n<td data-valign=\"top\" data-align=\"left\">7 to the second power or<\/td>\n<td data-valign=\"top\" data-align=\"left\">7 squared<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">5<sup>3<\/sup><\/td>\n<td data-valign=\"top\" data-align=\"left\">5 to the third power or<\/td>\n<td data-valign=\"top\" data-align=\"left\">5 cubed<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">9<sup>4<\/sup><\/td>\n<td data-valign=\"top\" data-align=\"left\">9 to the fourth power<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">12<sup>5<\/sup><\/td>\n<td data-valign=\"top\" data-align=\"left\">12 to the fifth power<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836416285\">\n<h3 data-type=\"title\">Simplify Expressions Using the Order of Operations<\/h3>\n<p id=\"fs-id1167836416290\">To <span data-type=\"term\">simplify an expression<\/span> means to do all the math possible. For example, to simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06e49f3b6bd8891b43b7c5f52210131f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&middot;&#50;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"47\" style=\"vertical-align: -2px;\" \/> we would first multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5b3f3e3efd93b151dc0b8270ce754e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&middot;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836507828\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ecf0ebc5d48eaca887a9f57c07fa99c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&middot;&#50;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"47\" style=\"vertical-align: -22px;\" \/><\/div>\n<p id=\"fs-id1167836534471\">By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836534475\">\n<div data-type=\"title\">Simplify an Expression<\/div>\n<p id=\"fs-id1167836534481\">To <strong data-effect=\"bold\">simplify an expression<\/strong>, do all operations in the expression.<\/p>\n<\/div>\n<p id=\"fs-id1167836534490\">We\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the <span data-type=\"term\">order of operations<\/span>. Otherwise, expressions may have different meanings, and they may result in different values.<\/p>\n<p id=\"fs-id1167836525166\">For example, consider the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17d17e06fa864065b9b1fc0e92cd9c42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#43;&#51;&middot;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"53\" style=\"vertical-align: -2px;\" \/> Some students simplify this getting 49, by adding <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ddef8037f7e2d96fcb215248d42b9190_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/> and then multiplying that result by 7. Others get 25, by multiplying <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e92f94ab8e94d7257e92a56d8b198f28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&middot;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> first and then adding 4.<\/p>\n<p id=\"fs-id1167836522816\">The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836522820\" class=\"howto\">\n<div data-type=\"title\">Use the order of operations.<\/div>\n<ol id=\"fs-id1167836522828\" type=\"1\" class=\"stepwise\">\n<li>Parentheses and Other Grouping Symbols\n<ul id=\"fs-id1167829590631\" data-bullet-style=\"bullet\">\n<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\n<\/ul>\n<\/li>\n<li>Exponents\n<ul id=\"fs-id1167829590645\" data-bullet-style=\"bullet\">\n<li>Simplify all expressions with exponents.<\/li>\n<\/ul>\n<\/li>\n<li>Multiplication and Division\n<ul id=\"fs-id1167836375957\" data-bullet-style=\"bullet\">\n<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<\/li>\n<li>Addition and Subtraction\n<ul id=\"fs-id1167836375971\" data-bullet-style=\"bullet\">\n<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836375984\">Students often ask, \u201cHow will I remember the order?\u201d Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase \u201cPlease Excuse My Dear Aunt Sally\u201d.<\/p>\n<div data-type=\"equation\" id=\"fs-id1166502309552\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa964036da889d26da57fb2a7585787_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#114;&#101;&#110;&#116;&#104;&#101;&#115;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#101;&#97;&#115;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#99;&#117;&#115;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#108;&#116;&#105;&#112;&#108;&#105;&#99;&#97;&#116;&#105;&#111;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#118;&#105;&#115;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#97;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#100;&#105;&#116;&#105;&#111;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#110;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#108;&#108;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"395\" style=\"vertical-align: -36px;\" \/><\/div>\n<p id=\"fs-id1167836619914\">It\u2019s good that \u201c<strong data-effect=\"bold\">M<\/strong>y <strong data-effect=\"bold\">D<\/strong>ear\u201d goes together, as this reminds us that <strong data-effect=\"bold\">m<\/strong>ultiplication and <strong data-effect=\"bold\">d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.<\/p>\n<p id=\"fs-id1167829597218\">Similarly, \u201c<strong data-effect=\"bold\">A<\/strong>unt <strong data-effect=\"bold\">S<\/strong>ally\u201d goes together and so reminds us that <strong data-effect=\"bold\">a<\/strong>ddition and <strong data-effect=\"bold\">s<\/strong>ubtraction also have equal priority and we do them in order from left to right.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836508031\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836508033\">\n<div data-type=\"problem\" id=\"fs-id1167836508035\">\n<p id=\"fs-id1167836508037\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-341d9ec2fd7fb3bf8151e3947a71f0f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&divide;&#54;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829692917\">\n<table id=\"fs-id1167829692920\" class=\"unnumbered unstyled can-break\" summary=\"The expression is 18 divided by 6 plus 4 open parentheses 5 minus 2 close parentheses. Since there are parentheses, we first open them by performing the subtraction 5 minus 2. The expression now is 18 divided by 6 plus 4 times 3. There are no exponents. Next we check for multiplication and division. Divide first because we multiply and divide left to right. We now have 3 plus 4 times 3. Next we multiply. We now have 3 plus 12. There is no other multiplication or division. Finally, we check for addition or subtraction. We add to get the number 15.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829596519\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Parentheses? Yes, subtract first.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829596546\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide first because we multiply and divide left to right.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836508078\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Any other multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829695213\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Any other multiplication of division? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Any addition or subtraction? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836531883\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_009e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836698551\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836698555\">\n<div data-type=\"problem\" id=\"fs-id1167836698557\">\n<p id=\"fs-id1167836698559\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f3d50874c99581f1aaaa47db3cffbb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&divide;&#53;&#43;&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836627072\">\n<p id=\"fs-id1167836627074\">16<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836627080\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836627083\">\n<div data-type=\"problem\" id=\"fs-id1167836627085\">\n<p id=\"fs-id1167836627088\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4602fe6cc4ca61216f8dcab2b3db771e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#48;&divide;&#49;&#48;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829586618\">\n<p id=\"fs-id1167829586620\">23<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829586626\">When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829586631\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829586633\">\n<div data-type=\"problem\" id=\"fs-id1167829586635\">\n<p id=\"fs-id1167829586637\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf039663e37e802b284797c27639889d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#91;&#54;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833056784\">\n<table id=\"fs-id1167833056787\" class=\"unnumbered unstyled can-break\" summary=\"The expression is 5 plus 2 to the power 3 plus 3 open bracket 6 minus 3 open parentheses 4 minus 2 close parentheses close bracket. Focus on the parentheses that are inside the brackets. Subtract to get 5 plus 2 to the power 3 plus 3 open bracket 6 minus 3 open parentheses 2 close parentheses close bracket. Continue inside the brackets and multiply to get 5 plus 2 to the power 3 plus 3 open bracket 6 minus 6 close bracket. Continue inside the brackets and subtract to get 5 plus 2 to the power 3 plus 3 open bracket 0 close bracket. The expression inside the brackets requires no further simplification. Now simplify exponents to get 5 plus 8 plus 3 open bracket 0 close bracket. Check for multiplication or division. Multiply to get 5 plus 8 plus 0. Check for addition or subtraction. Finally add to get 13.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829694073\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Are there any parentheses (or other<\/p>\n<div data-type=\"newline\"><\/div>\n<p>grouping symbols)? Yes.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829694118\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Focus on the parentheses that are inside the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>brackets. Subtract.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836418416\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Continue inside the brackets and multiply.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836418442\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Continue inside the brackets and subtract.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836533805\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The expression inside the brackets requires<\/p>\n<div data-type=\"newline\"><\/div>\n<p>no further simplification.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Are there any exponents? Yes. Simplify exponents.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836533832\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Is there any multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836627134\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Is there any addition of subtraction? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836627175\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829693866\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_010j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829693883\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829693887\">\n<div data-type=\"problem\" id=\"fs-id1167829693889\">\n<p id=\"fs-id1167829693891\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4032a52ae4c783b4e2a22f2d2a023847_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#43;&#123;&#53;&#125;&#94;&#123;&#51;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#91;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"151\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836552370\">\n<p id=\"fs-id1167836552372\">86<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836552378\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836552382\">\n<div data-type=\"problem\" id=\"fs-id1167836552385\">\n<p id=\"fs-id1167836552387\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f1f5bce659bd285283b84e5f26e6e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#55;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"132\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836552426\">\n<p id=\"fs-id1167836552428\">1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836530230\">\n<h3 data-type=\"title\">Evaluate an Expression<\/h3>\n<p id=\"fs-id1167836530236\">In the last few examples, we simplified expressions using the order of operations. Now we\u2019ll evaluate some expressions\u2014again following the order of operations. To <span data-type=\"term\">evaluate an expression<\/span> means to find the value of the expression when the variable is replaced by a given number.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836530245\">\n<div data-type=\"title\">Evaluate an Expression<\/div>\n<p id=\"fs-id1167836530250\">To <strong data-effect=\"bold\">evaluate an expression<\/strong> means to find the value of the expression when the variable is replaced by a given number.<\/p>\n<\/div>\n<p id=\"fs-id1167836530261\">To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836530265\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836530267\">\n<div data-type=\"problem\" id=\"fs-id1167836530269\">\n<p id=\"fs-id1167836530271\">Evaluate when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-914939b505e8fbb8d2291b61e8ddacd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d75b22bd2753a2e9cffa727e9fc19cdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87c63bc7558aae8ac7c50c146b3d8c43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836512101\">\n<p id=\"fs-id1171791331671\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836512118\" class=\"unnumbered unstyled\" summary=\"The expression is x squared. Replace x with 4 to get 4 squared. Use definition of exponent to get 4 times 4. Simplify to get 16.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416388\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836416408\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416422\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use definition of exponent.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416449\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416476\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_011e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829595190\" class=\"unnumbered unstyled\" summary=\"The expression is 3 raised to the power x. Replace x with 4 to get 3 to the power 4. Use definition of exponent to get 3 times 3 times 3 times 3. Simplify to get 81.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829595233\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829595253\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829595267\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use definition of exponent.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836556778\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836556805\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_012e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836556826\" class=\"unnumbered unstyled\" summary=\"The expression is 2 x squared plus 3 x plus 8. Substitute x with 4 to get 2 open parentheses 4 close parentheses squared plus 3 open parentheses 4 close parentheses plus 8. Follow order of operations to first get 2 open parentheses 16 close parentheses plus 3 open parentheses 4 close parentheses plus 8. Then, 32 plus 12 plus 8. Then, 52.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836556871\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836556890\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571162\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Follow the order of operations.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571189\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571214\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836571240\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_013f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836571258\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836571262\">\n<div data-type=\"problem\" id=\"fs-id1167836571264\">\n<p id=\"fs-id1167836571266\">Evaluate when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d62a78a97a33b2bfba1ddcfe9dbb8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93acbaf8db86b4104d48da88bf046691_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5792b8388ad6d3fffe04658e940060a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836627717\">\n<p id=\"fs-id1167836627719\"><span class=\"token\">\u24d0<\/span> 9 <span class=\"token\">\u24d1<\/span> 64 <span class=\"token\">\u24d2<\/span>40<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836627740\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836627744\">\n<div data-type=\"problem\" id=\"fs-id1167836627747\">\n<p id=\"fs-id1167836627749\">Evaluate when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9892e673d9c8e401362276dfe3b69c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29a18debcfeba0c22f81de96b6c65e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d138fb64cc6a9225d06fdb1af9ca3a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48d39b81c72354e3f13eab799ba4104f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836627820\">\n<p id=\"fs-id1167836627822\"><span class=\"token\">\u24d0<\/span> 216 <span class=\"token\">\u24d1<\/span> 64 <span class=\"token\">\u24d2<\/span> 185<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836627844\">\n<h3 data-type=\"title\">Identify and Combine Like Terms<\/h3>\n<p id=\"fs-id1167836627849\">Algebraic expressions are made up of terms. A <span data-type=\"term\">term<\/span> is a constant, or the product of a constant and one or more variables.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836627856\">\n<div data-type=\"title\">Term<\/div>\n<p id=\"fs-id1167836627861\">A <strong data-effect=\"bold\">term<\/strong> is a constant or the product of a constant and one or more variables.<\/p>\n<\/div>\n<p id=\"fs-id1167836448104\">Examples of terms are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98a19059eae606860c6c2b28516e37ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#44;&#121;&#44;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#57;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c8b1f68467aa97444acbc524e9d7754_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167836448144\">The constant that multiplies the variable is called the <span data-type=\"term\">coefficient<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836448151\">\n<div data-type=\"title\">Coefficient<\/div>\n<p id=\"fs-id1167836448156\">The <strong data-effect=\"bold\">coefficient<\/strong> of a term is the constant that multiplies the variable in a term.<\/p>\n<\/div>\n<p id=\"fs-id1167836448165\">Think of the coefficient as the number in front of the variable. The coefficient of the term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcea841b93e6d1c6150bf94b4036ab3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> is 3. When we write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> the coefficient is 1, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10efd1e547ae62ff7a76805a54cfcc12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&middot;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167836448199\">Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are <span data-type=\"term\">like terms<\/span>.<\/p>\n<p id=\"fs-id1167836448207\">Look at the following 6 terms. Which ones seem to have traits in common?<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836448210\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e58127bab2bcee7b70e4fe86b6c35f78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"190\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"fs-id1167836448258\">We say,<\/p>\n<p id=\"fs-id1167836448262\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e05d6a7b1f3cb1e28c97922d49af047d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/> are like terms.<\/p>\n<p id=\"fs-id1167836652453\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbd7c54b5ed6d3514d4f0f3a1711ff25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcea841b93e6d1c6150bf94b4036ab3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> are like terms.<\/p>\n<p id=\"fs-id1167836652473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-240345e4074d649d87a911f594810a9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25f8f5b2e125247b58f799d23dc7005f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/> are like terms.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836652497\">\n<div data-type=\"title\">Like Terms<\/div>\n<p id=\"fs-id1167836652502\">Terms that are either constants or have the same variables raised to the same powers are called <strong data-effect=\"bold\">like terms.<\/strong><\/p>\n<\/div>\n<p id=\"fs-id1167836652512\">If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836652516\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1fc720318b79d0dee6f889ad4a19b37e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#120;&#43;&#55;&#120;&#43;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#101;&#102;&#102;&#105;&#99;&#105;&#101;&#110;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"369\" style=\"vertical-align: -12px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1167836652573\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How To Combine Like Terms<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836652578\">\n<div data-type=\"problem\" id=\"fs-id1167836652581\">\n<p id=\"fs-id1167836652583\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d69471151f89b76680eee0c40e5fa9f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#55;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"213\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836611286\"><span data-type=\"media\" id=\"fs-id1167836611288\" data-alt=\"Step 1 is to identify the like terms in 2 x squared plus 3 x plus 7 plus x squared plus 4 x plus 5. The like terms are 2 x squared and x squared, then 3 x and 4 x, then 7 and 5.\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_01_01_014_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to identify the like terms in 2 x squared plus 3 x plus 7 plus x squared plus 4 x plus 5. The like terms are 2 x squared and x squared, then 3 x and 4 x, then 7 and 5.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836611301\" data-alt=\"Step 2 is to rearrange the expression so the like terms are together. Hence, we have 2 x squared plus x squared plus 3 x plus 4 x plus 7 plus 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_014b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rearrange the expression so the like terms are together. Hence, we have 2 x squared plus x squared plus 3 x plus 4 x plus 7 plus 5.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836611314\" data-alt=\"Step 3 is to combine the like terms to get 3 x squared plus 7 x plus 12.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_014c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to combine the like terms to get 3 x squared plus 7 x plus 12.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836611328\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836611332\">\n<div data-type=\"problem\" id=\"fs-id1167836611335\">\n<p id=\"fs-id1167836611337\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14bdf021254067e0a22ec3d23c9f3def_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#57;&#43;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"222\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836611384\">\n<p id=\"fs-id1167836611386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8943ca217f24433d71f5869ad37f952e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#120;&#43;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836611411\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836611416\">\n<div data-type=\"problem\" id=\"fs-id1167836611418\">\n<p id=\"fs-id1167836611420\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9859728652693690c9552703b2e8c6aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#121;&#43;&#50;&#43;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836792868\">\n<p id=\"fs-id1167836792871\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-539ff4e0d17ba9baf9f35b31cc001e84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#121;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836792896\" class=\"howto\">\n<div data-type=\"title\">Combine like terms.<\/div>\n<ol id=\"fs-id1167836792904\" type=\"1\" class=\"stepwise\">\n<li>Identify like terms.<\/li>\n<li>Rearrange the expression so like terms are together.<\/li>\n<li>Add or subtract the coefficients and keep the same variable for each group of like terms.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836792925\">\n<h3 data-type=\"title\">Translate an English Phrase to an Algebraic Expression<\/h3>\n<p id=\"fs-id1167836792931\">We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we\u2019ve talked about will help us do that. <a href=\"#fs-id1167836792941\" class=\"autogenerated-content\">(Figure)<\/a> summarizes them.<\/p>\n<table id=\"fs-id1167836792941\" summary=\"This table has three columns labeled operation, phrase and expression. There are four rows. The phrases for addition are a plus b, the sum of a and b, a increased by b, the total of a and b, b added to a. The expression is a plus b. The phrases for subtraction are a minus b, the difference of a and b, a decreased by b, b less than a, b subtracted from a. The expression is a minus b. The phrases for multiplication are a times b, the product of a and b, 2a. The expressions are a dot b, ab, a open parentheses b close parentheses, open parentheses a parentheses open parentheses b close parentheses and 2a. The phrases for division are a divided by b, the quotient of a and b, the ratio of a and b, b divided into a. The expressions are a divided by b, a slash b, a upon b, b parentheses a overbar.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"left\">Operation<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Phrase<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Addition<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> plus <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">a<\/em> increased by <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> more than <em data-effect=\"italics\">a<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the total of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> added to <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38830949a60ce6786a6fdf6309482002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Subtraction<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> minus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the difference of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">a<\/em> decreased by <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> less than <em data-effect=\"italics\">a<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> subtracted from <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b644d18119c5701b308f883bd50656d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Multiplication<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> times <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>twice <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69ca78931821fcafa666a388284aa108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;&#44;&#97;&#98;&#44;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02baaffe4ddb6a44400eb7ba175e566c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Division<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> divided by <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the quotient of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the ratio of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> divided into <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-748dadb6282b269320bc3f4d463478b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&divide;&#98;&#44;&#97;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#98;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#44;&#98;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"103\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836513163\">Look closely at these phrases using the four operations:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836513166\" data-alt=\"The sum of a and b, the difference of a and b, the product of a and b, the quotient of a and b.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_015_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The sum of a and b, the difference of a and b, the product of a and b, the quotient of a and b.\" \/><\/span><\/p>\n<p id=\"fs-id1167836513178\">Each phrase tells us to operate on two numbers. Look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to find the numbers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836513192\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836513194\">\n<div data-type=\"problem\" id=\"fs-id1167836513196\">\n<p id=\"fs-id1167836513198\">Translate each English phrase into an algebraic expression:<\/p>\n<p id=\"fs-id1167836513201\"><span class=\"token\">\u24d0<\/span> the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0248a072c537cefc7e0c041b8f18b50d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -1px;\" \/> and 9<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the quotient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-649c835f6f65cf7f7c974a270b0c159b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -4px;\" \/> and 3<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> twelve more than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> seven less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db83d2a5841cc9737e8347844484d1b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836520411\">\n<p id=\"fs-id1167836520413\"><span class=\"token\">\u24d0<\/span> The key word is <em data-effect=\"italics\">difference<\/em>, which tells us the operation is subtraction. Look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and t<\/em>o find the numbers to subtract.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836520436\" data-alt=\"The difference of 14 x and 9, 14 x minus 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_016_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The difference of 14 x and 9, 14 x minus 9.\" \/><\/span><\/p>\n<p id=\"fs-id1167836520448\"><span class=\"token\">\u24d1<\/span> The key word is <em data-effect=\"italics\">quotient<\/em>, which tells us the operation is division.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836520460\" data-alt=\"The quotient of 8 y squared and 3, divide 8 y squared by 3, 8 y squared divided by 3. This can also be written as 8 y squared slash 3 or 8 y squared upon 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_017_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The quotient of 8 y squared and 3, divide 8 y squared by 3, 8 y squared divided by 3. This can also be written as 8 y squared slash 3 or 8 y squared upon 3.\" \/><\/span><\/p>\n<p id=\"fs-id1167836520473\"><span class=\"token\">\u24d2<\/span> The key words are <em data-effect=\"italics\">more than.<\/em> They tell us the operation is addition. <em data-effect=\"italics\">More than<\/em> means \u201cadded to.\u201d<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836520491\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27bd6649fdd1f381376278007883ba00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#119;&#101;&#108;&#118;&#101;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#119;&#101;&#108;&#118;&#101;&#32;&#97;&#100;&#100;&#101;&#100;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#43;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"147\" style=\"vertical-align: -26px;\" \/><\/div>\n<p id=\"fs-id1167836520533\"><span class=\"token\">\u24d3<\/span> The key words are <em data-effect=\"italics\">less than<\/em>. They tell us to subtract. <em data-effect=\"italics\">Less than<\/em> means \u201csubtracted from.\u201d<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836520550\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d70fbd6e1d2beec8ff29778747a3c0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#118;&#101;&#110;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#118;&#101;&#110;&#32;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#101;&#100;&#32;&#102;&#114;&#111;&#109;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"210\" style=\"vertical-align: -23px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836520235\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836520239\">\n<div data-type=\"problem\" id=\"fs-id1167836520241\">\n<p id=\"fs-id1167836520243\">Translate the English phrase into an algebraic expression:<\/p>\n<p id=\"fs-id1167836520246\"><span class=\"token\">\u24d0<\/span> the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54501ffd336bc114d5e7e5b41a8d7baf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" style=\"vertical-align: -1px;\" \/> and 13<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the quotient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cc37a51efcd4650021f5d9d8222e893_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -1px;\" \/> and 2<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> 13 more than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> 18 less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2e766fef3082260b54b584383d424b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836520301\">\n<p id=\"fs-id1167836520303\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a8d27d55d4b18f5f26d6fd04637f8da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed43c25a44822fa0b74542a871eab018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#120;&divide;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-635f4ad82fb997b47077f1c4e2b07e40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#43;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-baba7410167b4903e0ef80d23c0cfbd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836513469\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836513472\">\n<div data-type=\"problem\" id=\"fs-id1167836513474\">\n<p id=\"fs-id1167836513476\">Translate the English phrase into an algebraic expression:<\/p>\n<p id=\"fs-id1167836513480\"><span class=\"token\">\u24d0<\/span> the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3857e4bc2f103dae13382cdbfa436d3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"33\" style=\"vertical-align: -4px;\" \/> and 19<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aca3387e59afe477960eabc2f23b3db5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <em data-effect=\"italics\">y<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Eleven more than <em data-effect=\"italics\">x<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Fourteen less than 11<em data-effect=\"italics\">a<\/em><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836513534\">\n<p id=\"fs-id1167836513536\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15593485fd763192f03067ba6f631c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-190eea1fe2fb9d508969d8a6808f6a1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-339028354620d2bf8b9638e51b7c9347_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7e6cfee946fe60d0fbcd60638597ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#97;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1171790448010\">We look carefully at the words to help us distinguish between multiplying a sum and adding a product.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836513605\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836513607\">\n<div data-type=\"problem\" id=\"fs-id1167836513609\">\n<p id=\"fs-id1167836513612\">Translate the English phrase into an algebraic expression:<\/p>\n<p id=\"fs-id1167836513615\"><span class=\"token\">\u24d0<\/span> eight times the sum of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the sum of eight times <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836512532\">\n<p id=\"fs-id1167836512534\">There are two operation words\u2014<em data-effect=\"italics\">times<\/em> tells us to multiply and <em data-effect=\"italics\">sum<\/em> tells us to add.<\/p>\n<p id=\"fs-id1167836512547\"><span class=\"token\">\u24d0<\/span> Because we are multiplying 8 times the sum, we need parentheses around the sum of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcbce862c7ce88950b44e44e510fa8c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/> This forces us to determine the sum first. (Remember the order of operations.)<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836512583\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39a3c55a61353379ce0f0f5039e682be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#105;&#103;&#104;&#116;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#115;&#117;&#109;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"232\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"fs-id1167836512634\"><span class=\"token\">\u24d1<\/span> To take a sum, we look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to see what is being added. Here we are taking the sum <em data-effect=\"italics\">of<\/em> eight times <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836512668\" data-alt=\"The sum of 8 times x and y is 8 x plus y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_018_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The sum of 8 times x and y is 8 x plus y.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836620952\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836620957\">\n<div data-type=\"problem\" id=\"fs-id1167836620959\">\n<p id=\"fs-id1167836620961\">Translate the English phrase into an algebraic expression:<\/p>\n<p id=\"fs-id1167836620964\"><span class=\"token\">\u24d0<\/span> four times the sum of <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the sum of four times <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836620997\">\n<p id=\"fs-id1167836621000\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6311bed1436cbe0038471fde3ac42e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7626499e51cccbc86dead204e5bb82b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#112;&#43;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836621044\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836621048\">\n<div data-type=\"problem\" id=\"fs-id1167836621050\">\n<p id=\"fs-id1167836621052\">Translate the English phrase into an algebraic expression:<\/p>\n<p id=\"fs-id1167836621056\"><span class=\"token\">\u24d0<\/span> the difference of two times <em data-effect=\"italics\">x<\/em> and 8<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> two times the difference of <em data-effect=\"italics\">x<\/em> and 8<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836621081\">\n<p id=\"fs-id1167836621083\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40fd682e49e250ca37ef17c4e9e591d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1fa1b77ccb4e07e066e4248c48fa81df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836606926\">Later in this course, we\u2019ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We\u2019ll see how to do this in the next two examples.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836606933\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836606935\">\n<div data-type=\"problem\" id=\"fs-id1167836606937\">\n<p id=\"fs-id1167836606939\">The length of a rectangle is 14 less than the width. Let <em data-effect=\"italics\">w<\/em> represent the width of the rectangle. Write an expression for the length of the rectangle.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836606950\">\n<p id=\"fs-id1167836606952\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53ce8c586297240943e77898dac01432_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#32;&#112;&#104;&#114;&#97;&#115;&#101;&#32;&#97;&#98;&#111;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#99;&#116;&#97;&#110;&#103;&#108;&#101;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#32;&#116;&#104;&#101;&#32;&#119;&#105;&#100;&#116;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#119;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#32;&#96;&#96;&#116;&#104;&#101;&#32;&#119;&#105;&#100;&#116;&#104;&#46;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#119;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#101;&#100;&#32;&#102;&#114;&#111;&#109;&#125;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#32;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#101;&#100;&#32;&#102;&#114;&#111;&#109;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#119;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#112;&#104;&#114;&#97;&#115;&#101;&#32;&#105;&#110;&#116;&#111;&#32;&#97;&#108;&#103;&#101;&#98;&#114;&#97;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#119;&#45;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"692\" style=\"vertical-align: -36px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836607056\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836607060\">\n<div data-type=\"problem\" id=\"fs-id1167836607063\">\n<p id=\"fs-id1167836607065\">The length of a rectangle is 7 less than the width. Let <em data-effect=\"italics\">w<\/em> represent the width of the rectangle. Write an expression for the length of the rectangle.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836607075\">\n<p id=\"fs-id1167836607077\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c171cd55be2846043de71e9ec80741db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829696780\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829696784\">\n<div data-type=\"problem\" id=\"fs-id1167829696786\">\n<p id=\"fs-id1167829696788\">The width of a rectangle is 6 less than the length. Let <em data-effect=\"italics\">l<\/em> represent the length of the rectangle. Write an expression for the width of the rectangle.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829696798\">\n<p id=\"fs-id1167829696801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-642737f6dffabab57e18998f40ca4cc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1171790738586\">The expressions in the next example will be used in the typical coin mixture problems we will see soon.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829696814\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829696817\">\n<div data-type=\"problem\" id=\"fs-id1167829696819\">\n<p id=\"fs-id1167829696821\">June has dimes and quarters in her purse. The number of dimes is seven less than four times the number of quarters. Let <em data-effect=\"italics\">q<\/em> represent the number of quarters. Write an expression for the number of dimes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829696832\">\n<p id=\"fs-id1167829696834\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c911d795332761d6233a1ed1ae250d4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#32;&#112;&#104;&#114;&#97;&#115;&#101;&#32;&#97;&#98;&#111;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#100;&#105;&#109;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#118;&#101;&#110;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#32;&#102;&#111;&#117;&#114;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#113;&#117;&#97;&#114;&#116;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#113;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#113;&#117;&#97;&#114;&#116;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#32;&#52;&#32;&#116;&#105;&#109;&#101;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#113;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#32;&#52;&#32;&#116;&#105;&#109;&#101;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#113;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#32;&#52;&#125;&#113;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#112;&#104;&#114;&#97;&#115;&#101;&#32;&#105;&#110;&#116;&#111;&#32;&#97;&#108;&#103;&#101;&#98;&#114;&#97;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#113;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"127\" width=\"769\" style=\"vertical-align: -59px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829696930\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829696934\">\n<div data-type=\"problem\" id=\"fs-id1167829696936\">\n<p id=\"fs-id1167829696939\">Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let <em data-effect=\"italics\">q<\/em> represent the number of quarters. Write an expression for the number of dimes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836511104\">\n<p id=\"fs-id1167836511106\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db433ed80a5b5325d550839ff053a5bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#113;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836511122\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836511126\">\n<div data-type=\"problem\" id=\"fs-id1167836511128\">\n<p id=\"fs-id1167836511130\">Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let <em data-effect=\"italics\">n<\/em> represent the number of nickels. Write an expression for the number of dimes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836511141\">\n<p id=\"fs-id1167836511143\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ebc4f0e2a5d562eb99e9bf01ae6e372_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#110;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836511160\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836511167\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Divisibility Tests<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> A number is divisible by:<\/p>\n<div data-type=\"newline\"><\/div>\n<p> \u2003\u20032 if the last digit is 0, 2, 4, 6, or 8.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> \u2003\u20033 if the sum of the digits is divisible by 3.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> \u2003\u20035 if the last digit is 5 or 0.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> \u2003\u20036 if it is divisible by both 2 and 3.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> \u2003\u200310 if it ends with 0.<\/li>\n<li><strong data-effect=\"bold\">How to find the prime factorization of a composite number.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167836511200\" type=\"1\" class=\"stepwise\">\n<li>Find two factors whose product is the given number, and use these numbers to create two branches.<\/li>\n<li>If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.<\/li>\n<li>If a factor is not prime, write it as the product of two factors and continue the process.<\/li>\n<li>Write the composite number as the product of all the circled primes.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How To Find the least common multiple using the prime factors method.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167836511232\" type=\"1\" class=\"stepwise\">\n<li>Write each number as a product of primes.<\/li>\n<li>List the primes of each number. Match primes vertically when possible.<\/li>\n<li>Bring down the columns.<\/li>\n<li>Multiply the factors.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Equality Symbol<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e69295ccf8d8ca292abcc97f861e345f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/> is read \u201c<em data-effect=\"italics\">a<\/em> is equal to <em data-effect=\"italics\">b<\/em>.\u201d<\/p>\n<div data-type=\"newline\"><\/div>\n<p> The symbol \u201c=\u201d is called the equal sign.<\/li>\n<li><strong data-effect=\"bold\">Inequality<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831871677\" data-alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_020_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">Inequality Symbols<\/strong>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836519850\" summary=\"The table describes inequality symbols in words. The symbols described are a is not equal to b, a is less than b, a is less than or equal to b, a is greater then b, a is greater than or equal to b.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Inequality Symbols<\/th>\n<th data-valign=\"top\" data-align=\"left\">Words<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8740efbef2c653ea7d353859ecc91afd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">not equal to b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b57677a3c7d31d00aea3cdd09443b09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22bb7f496e0a4516393886afb53ffa7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">less than or equal to b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b84b1480aef6484626cffaeccec0b9fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than b.<\/em><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4925aae9d8237338b0824392ba2d182a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#103;&#101;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> is <em data-effect=\"italics\">greater than or equal to b.<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><strong data-effect=\"bold\">Grouping Symbols<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4909a61294f5368dc2c706f498146843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#114;&#101;&#110;&#116;&#104;&#101;&#115;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#107;&#101;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"193\" style=\"vertical-align: -27px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Exponential Notation<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef78644f25c125ead81d3c69ef32dee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> means multiply <em data-effect=\"italics\">a<\/em> by itself, <em data-effect=\"italics\">n<\/em> times.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> The expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef78644f25c125ead81d3c69ef32dee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> is read <em data-effect=\"italics\">a<\/em> to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02553ae49d720703c253c7bcbf008617_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#116;&#104;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"24\" style=\"vertical-align: 0px;\" \/> power.<\/li>\n<li><strong data-effect=\"bold\">Simplify an Expression<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> To simplify an expression, do all operations in the expression.<\/li>\n<li><strong data-effect=\"bold\">How to use the order of operations.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167836545484\" type=\"1\" class=\"stepwise\">\n<li>Parentheses and Other Grouping Symbols\n<ul id=\"fs-id1167836545494\" data-bullet-style=\"bullet\">\n<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\n<\/ul>\n<\/li>\n<li>Exponents\n<ul id=\"fs-id1167836545508\" data-bullet-style=\"bullet\">\n<li>Simplify all expressions with exponents.<\/li>\n<\/ul>\n<\/li>\n<li>Multiplication and Division\n<ul id=\"fs-id1167836545520\" data-bullet-style=\"bullet\">\n<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<\/li>\n<li>Addition and Subtraction\n<ul id=\"fs-id1167836545534\" data-bullet-style=\"bullet\">\n<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to combine like terms.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167836545552\" type=\"1\" class=\"stepwise\">\n<li>Identify like terms.<\/li>\n<li>Rearrange the expression so like terms are together.<\/li>\n<li>Add or subtract the coefficients and keep the same variable for each group of like terms.<\/li>\n<\/ol>\n<table id=\"fs-id1167836545572\" summary=\"This table has three columns labeled operation, phrase and expression. There are four rows. The phrases for addition are a plus b, the sum of a and b, a increased by b, the total of a and b, b added to a. The expression is a plus b. The phrases for subtraction are a minus b, the difference of a and b, a decreased by b, b less than a, b subtracted from a. The expression is a minus b. The phrases for multiplication are a times b, the product of a and b, 2a. The expressions are a dot b, ab, a open parentheses b close parentheses, open parentheses a parentheses open parentheses b close parentheses and 2a. The phrases for division are a divided by b, the quotient of a and b, the ratio of a and b, b divided into a. The expressions are a divided by b, a slash b, a upon b, b parentheses a overbar.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"left\">Operation<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Phrase<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Addition<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> plus <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">a<\/em> increased by <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> more than <em data-effect=\"italics\">a<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the total of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> added to <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38830949a60ce6786a6fdf6309482002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Subtraction<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> minus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the difference of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">a<\/em> decreased by <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> less than <em data-effect=\"italics\">a<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> subtracted from <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b644d18119c5701b308f883bd50656d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Multiplication<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> times <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>twice <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69ca78931821fcafa666a388284aa108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;&#44;&#97;&#98;&#44;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02baaffe4ddb6a44400eb7ba175e566c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Division<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> divided by <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>the quotient of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>the ratio of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> divided into <em data-effect=\"italics\">a<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-748dadb6282b269320bc3f4d463478b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&divide;&#98;&#44;&#97;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#98;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#44;&#98;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"103\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829693720\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829693724\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167829693732\"><strong data-effect=\"bold\">Identify Multiples and Factors<\/strong><\/p>\n<p id=\"fs-id1167829693738\">In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829693742\">\n<div data-type=\"problem\" id=\"fs-id1167829693745\">\n<p id=\"fs-id1167829693747\">84<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829693751\">\n<p id=\"fs-id1167829693753\">Divisible by 2, 3, 6<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829693758\">\n<div data-type=\"problem\" id=\"fs-id1167829693761\">\n<p id=\"fs-id1167829693763\">96<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829693774\">\n<div data-type=\"problem\" id=\"fs-id1167829693777\">\n<p id=\"fs-id1167829693779\">896<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829693783\">\n<p id=\"fs-id1167829693785\">Divisible by 2<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829693790\">\n<div data-type=\"problem\" id=\"fs-id1167829693793\">\n<p id=\"fs-id1167829693795\">942<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706817\">\n<div data-type=\"problem\" id=\"fs-id1167836706819\">\n<p id=\"fs-id1167836706821\">22,335<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836706825\">\n<p id=\"fs-id1167836706828\">Divisible by 3, 5<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706833\">\n<div data-type=\"problem\" id=\"fs-id1167836706835\">\n<p id=\"fs-id1167836706837\">39,075<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836706849\"><strong data-effect=\"bold\">Find Prime Factorizations and Least Common Multiples<\/strong><\/p>\n<p id=\"fs-id1167836706854\">In the following exercises, find the prime factorization.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836706857\">\n<div data-type=\"problem\" id=\"fs-id1167836706860\">\n<p id=\"fs-id1167836706862\">86<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836706866\">\n<p id=\"fs-id1167836706868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33ddc805cf5448344a44af462fbd66e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#52;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706881\">\n<div data-type=\"problem\" id=\"fs-id1167836706883\">\n<p id=\"fs-id1167836706885\">78<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706909\">\n<div data-type=\"problem\" id=\"fs-id1167836706911\">\n<p id=\"fs-id1167836706913\">455<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836706917\">\n<p id=\"fs-id1167836706919\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b78518fd8dc29ebca4255b1f534d5a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&middot;&#55;&middot;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"36\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706936\">\n<div data-type=\"problem\" id=\"fs-id1167836706938\">\n<p id=\"fs-id1167836706941\">400<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706977\">\n<div data-type=\"problem\" id=\"fs-id1167836706979\">\n<p id=\"fs-id1167836706981\">432<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836706985\">\n<p id=\"fs-id1167836706988\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a11b4429921f0c0fc0eb76c02a9516da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#50;&middot;&#50;&middot;&#51;&middot;&#51;&middot;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"63\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836579693\">\n<div data-type=\"problem\" id=\"fs-id1167836579695\">\n<p id=\"fs-id1167836579697\">627<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836579721\">In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836579725\">\n<div data-type=\"problem\" id=\"fs-id1167836579727\">\n<p id=\"fs-id1167836579729\">8, 12<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836579734\">\n<p id=\"fs-id1167836579736\">24<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836579741\">\n<div data-type=\"problem\" id=\"fs-id1167836579743\">\n<p id=\"fs-id1167836579745\">12, 16<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836579757\">\n<div data-type=\"problem\" id=\"fs-id1167836579759\">\n<p id=\"fs-id1167836579761\">28, 40<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836579766\">\n<p id=\"fs-id1167836579768\">420<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836579773\">\n<div data-type=\"problem\" id=\"fs-id1167836579775\">\n<p id=\"fs-id1167836579777\">84, 90<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836579789\">\n<div data-type=\"problem\" id=\"fs-id1167836579791\">\n<p id=\"fs-id1167836579793\">55, 88<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836579798\">\n<p id=\"fs-id1167836579800\">440<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836579805\">\n<div data-type=\"problem\" id=\"fs-id1167836579807\">\n<p id=\"fs-id1167836579809\">60, 72<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836579821\"><strong data-effect=\"bold\">Simplify Expressions Using the Order of Operations<\/strong><\/p>\n<p id=\"fs-id1167836579827\">In the following exercises, simplify each expression.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836579831\">\n<div data-type=\"problem\" id=\"fs-id1167836579833\">\n<p id=\"fs-id1167836579835\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93087f70711e4a6b7e0d6c22a1379e5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836692441\">\n<p id=\"fs-id1167836692443\">5<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836692448\">\n<div data-type=\"problem\" id=\"fs-id1167836692450\">\n<p id=\"fs-id1167836692452\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f81dad49d86f5429bed8d58c94aa520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836692492\">\n<div data-type=\"problem\" id=\"fs-id1167836692494\">\n<p id=\"fs-id1167836692496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1ad557dac274b0cecd3d0f2409d2881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836692522\">\n<p id=\"fs-id1167836692524\">58<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836692530\">\n<div data-type=\"problem\" id=\"fs-id1167836692532\">\n<p id=\"fs-id1167836692534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-568d3450e122c24e54da096592650583_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836692568\">\n<div data-type=\"problem\" id=\"fs-id1167836692570\">\n<p id=\"fs-id1167836692572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a2256556cead936144ce470ab077bfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&divide;&#52;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836620005\">\n<p id=\"fs-id1167836620007\">29<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836620012\">\n<div data-type=\"problem\" id=\"fs-id1167836620015\">\n<p id=\"fs-id1167836620017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a81f788c63a554b7f96944427a8b011_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#51;&divide;&#51;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836620053\">\n<div data-type=\"problem\" id=\"fs-id1167836620055\">\n<p id=\"fs-id1167836620057\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-769085fc2011dc517286c86e55eeb611_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#57;&middot;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#123;&#52;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836620091\">\n<p id=\"fs-id1167836620093\">149<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836620099\">\n<div data-type=\"problem\" id=\"fs-id1167836620101\">\n<p id=\"fs-id1167836620103\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fdbc45cc60442f9d2522cdceeeb08386_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#56;&middot;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#123;&#55;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836620144\">\n<div data-type=\"problem\" id=\"fs-id1167836620146\">\n<p id=\"fs-id1167836620148\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30e1bf8798a5926bd4107059d2e499ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#49;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836620182\">\n<p id=\"fs-id1167836620184\">50<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836685118\">\n<div data-type=\"problem\" id=\"fs-id1167836685120\">\n<p id=\"fs-id1167836685122\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-139b3764e45e3e513fa3cc4300f08ade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#91;&#50;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836685162\">\n<div data-type=\"problem\" id=\"fs-id1167836685164\">\n<p id=\"fs-id1167836685167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d06f353256ec761276d2fbd253efd07b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#55;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#45;&#123;&#51;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"185\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836685211\">\n<p id=\"fs-id1167836685214\">5<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836685219\">\n<div data-type=\"problem\" id=\"fs-id1167836685221\">\n<p id=\"fs-id1167836685223\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9df3be41cba3429edaad6f77c45f41ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#91;&#54;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#45;&#123;&#50;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836685275\"><strong data-effect=\"bold\">Evaluate an Expression<\/strong><\/p>\n<p id=\"fs-id1167836685282\">In the following exercises, evaluate the following expressions.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836685285\">\n<div data-type=\"problem\" id=\"fs-id1167836685287\">\n<p id=\"fs-id1167836685289\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e211b3d19a6898a4c9192f117c1fe08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ade694690dad929018ebadb1c3acd5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93acbaf8db86b4104d48da88bf046691_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-865441b7ce5e3f688fff413ca3d51ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833048379\">\n<p id=\"fs-id1167833048381\"><span class=\"token\">\u24d0<\/span> 64 <span class=\"token\">\u24d1<\/span> 16 <span class=\"token\">\u24d2<\/span> 7<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833048401\">\n<div data-type=\"problem\" id=\"fs-id1167833048403\">\n<p id=\"fs-id1167833048405\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d62a78a97a33b2bfba1ddcfe9dbb8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48541f82788d9b97a870606bd6257cd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4274d1f645d512397665bf1f91d005d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#53;&#125;&#94;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c43ecc89655d6145c78af6c754301f88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833048496\">\n<div data-type=\"problem\" id=\"fs-id1167833048498\">\n<p id=\"fs-id1167833048500\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ba123a7c4e45585ccfc113cc2eee507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#44;&#121;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a2178289668673e158859bb6f6ce0b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#121;&#45;&#55;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836526227\">\n<p id=\"fs-id1167836526229\">21<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836526235\">\n<div data-type=\"problem\" id=\"fs-id1167836526237\">\n<p id=\"fs-id1167836526239\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9eecae5b9ad825b9e9a518b0632943d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;&#121;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92faec7fbff95afccb97744d11b691d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#121;&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836526295\">\n<div data-type=\"problem\" id=\"fs-id1167836526298\">\n<p id=\"fs-id1167836526300\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92de696d4c1cc8af326ac46efba444b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#48;&#44;&#121;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7eeeec0e0ca641a3ceb7d5886a708aaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836526342\">\n<p id=\"fs-id1167836526344\">9<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836526350\">\n<div data-type=\"problem\" id=\"fs-id1167836526352\">\n<p id=\"fs-id1167836526354\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03e6ee8f57cb2a584ea24505087018d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#51;&#44;&#98;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84264a6e3af632c1d67d4907fa750268_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"54\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836543680\"><strong data-effect=\"bold\">Simplify Expressions by Combining Like Terms<\/strong><\/p>\n<p id=\"fs-id1167836543686\">In the following exercises, simplify the following expressions by combining like terms.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836543689\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836543691\">\n<p id=\"fs-id1167836543694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce580f628334889597da80e22b5dcaf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#43;&#50;&#43;&#51;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"121\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836543718\">\n<p id=\"fs-id1167836543720\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f269e37f062caae6ef340c6a3392a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836543735\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836543737\">\n<p id=\"fs-id1167836543739\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d0cfb26bec8f9ed9a969c5cd05b7fc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#121;&#43;&#53;&#43;&#50;&#121;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836543781\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836543783\">\n<p id=\"fs-id1167836543785\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94fd7ff2fed28d6603ecbf0b3baa8c82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#97;&#43;&#55;&#43;&#53;&#97;&#45;&#50;&#43;&#55;&#97;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"198\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836543820\">\n<p id=\"fs-id1167836543823\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3daa8944ac39e7212769d6aa5f7774d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;&#97;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688236\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836688238\">\n<p id=\"fs-id1167836688240\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-681e5760841a1f1ab0b53ca0f18243c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#99;&#43;&#52;&#43;&#54;&#99;&#45;&#51;&#43;&#57;&#99;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"184\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688288\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836688290\">\n<p id=\"fs-id1167836688292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b23833fcbb2421e1a371b318d5c028e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#49;&#49;&#43;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"243\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836688336\">\n<p id=\"fs-id1167836688338\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c0c567786fb75af914944017a31700a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#48;&#120;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688363\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836688365\">\n<p id=\"fs-id1167836688367\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3d1e3a8f2b8ba056d28af43350214d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#98;&#43;&#49;&#48;&#43;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#98;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"216\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836536429\"><strong data-effect=\"bold\">Translate an English Phrase to an Algebraic Expression<\/strong><\/p>\n<p id=\"fs-id1167836536435\">In the following exercises, translate the phrases into algebraic expressions.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836536439\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836536441\">\n<p id=\"fs-id1167836536443\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-baa0c10f3b651189378ee7bb755eaf94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a21284c7f098b67212f336007f0ccfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"28\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the quotient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-deb0cab3ab47baee4144a32b3bc3b2ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6093d29b7c98d4d53162882fe6703e10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Twenty-one more than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dbdb28b1297d32a21861bc74971aeae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d7259967dcc19f7e9fd0bb878df3f97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836536531\">\n<p id=\"fs-id1167836536533\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56cbc9c19a85ce992b2a04a030f0e020_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ffdb19c67d29130597537b421236b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#53;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3953f065dc404733a62f9a30f43d6f00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9e69589322165ddcce15d8672fa65af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836558971\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836558973\">\n<p id=\"fs-id1167836558975\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd5af49ca5a81fe06648e23b11bf3ad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d6ce0e75280543044083addcf65f90a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"28\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the quotient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf4d9750b63f2b4fa34879942e958cba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcea841b93e6d1c6150bf94b4036ab3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Eighteen more than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91fa2bda4c48f12d3beccdcacaba4770_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/>;<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d183d8dc1208efb574ffbc58c7d5aba3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"25\" style=\"vertical-align: -1px;\" \/> less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b8b34f8a8c82869bceb4690385c70a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833329552\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833329554\">\n<p id=\"fs-id1167833329556\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a70ae72d3800814ed3293500e3a415d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#97;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"33\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-156a656e95c6edb8e5afb0e6785e28b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"34\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27a6da6987a8c21eaa918b9915dc0b01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6093d29b7c98d4d53162882fe6703e10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Fifteen more than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4dc36d20a7690e84c99b1c91d651a08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae9131f15f6a30c1b2c451aea5d3353a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833329644\">\n<p id=\"fs-id1167833329646\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28674753e050d1ab50e20957cfc32123_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#97;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a091af9327d200e81c364e29656a9168_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc3122d2995f1cff81ac6e55bbf88a0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ad96bf937bf707132eecfb59a7ad7d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -1px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82c57082bf0859999c89396c9d98609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#120;&#60;&#49;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833412878\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833412880\">\n<p id=\"fs-id1167833412882\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e002f4676eb10e238e8487fce6b4b8ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"36\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04dcca9d1555a58330793cc4924ab091_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"35\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccbf485e4b3f7b0577f4c9fca4f15527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"35\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bfbbd0ada73774d5a760db14bc8fef1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Twelve more than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9608403b176cb023606ca01493d9f883_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56fc7ebd82b7bb8226aae7f1108c5256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/> less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b8012f4eeebd91ad845ec1a345d0610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"35\" style=\"vertical-align: 0px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832930223\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832930225\">\n<p id=\"fs-id1167832930227\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> eight times the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> and nine<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the difference of eight times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> and 9<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832930249\">\n<p id=\"fs-id1167832930251\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e28d1b60c29291b0b66928d13a880379_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d950e6f02cbff6ce0f4017631d2fe24e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#121;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832930295\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832930297\">\n<p id=\"fs-id1167832930299\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> seven times the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> and one<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the difference of seven times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> and 1<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832930367\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832930369\">\n<p id=\"fs-id1167832930371\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> five times the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcea841b93e6d1c6150bf94b4036ab3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the sum of five times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcea841b93e6d1c6150bf94b4036ab3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167832951106\">\n<p id=\"fs-id1167832951108\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5740390c9dbcb10950145921c62a5ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1990c03aea4a1c7c7f52dfe07d4d149_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#120;&#43;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"58\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832951153\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832951156\">\n<p id=\"fs-id1167832951158\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> eleven times the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06f7fb33ee72b9f6b38da946a36b77cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6093d29b7c98d4d53162882fe6703e10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> the sum of eleven times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06f7fb33ee72b9f6b38da946a36b77cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6093d29b7c98d4d53162882fe6703e10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832951265\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832951267\">\n<p id=\"fs-id1167832951269\">Eric has rock and country songs on his playlist. The number of rock songs is 14 more than twice the number of country songs. Let <em data-effect=\"italics\">c<\/em> represent the number of country songs. Write an expression for the number of rock songs.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832951281\">\n<p id=\"fs-id1167832951283\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2ba72f1d9dd53cb2816bc11ebdb9a14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#62;&#50;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518238\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836518240\">\n<p id=\"fs-id1167836518242\">The number of women in a Statistics class is 8 more than twice the number of men. Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> represent the number of men. Write an expression for the number of women.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518268\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836518270\">\n<p id=\"fs-id1167836518272\">Greg has nickels and pennies in his pocket. The number of pennies is seven less than three the number of nickels. Let <em data-effect=\"italics\">n<\/em> represent the number of nickels. Write an expression for the number of pennies.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518282\">\n<p id=\"fs-id1167836518284\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1aa24b15ed6f5e896b3c8d6346c47d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#110;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518299\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836518302\">\n<p id=\"fs-id1167836518304\">Jeannette has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8bb141885e33c3f53fa11e53e71f2af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-059afd90f56feeeeaeef7d574b9bf230_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> bills in her wallet. The number of fives is three more than six times the number of tens. Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> represent the number of tens. Write an expression for the number of fives.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836518346\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167836518354\">\n<div data-type=\"problem\" id=\"fs-id1167836518356\">\n<p id=\"fs-id1167836518358\">Explain in your own words how to find the prime factorization of a composite number.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518362\">\n<p id=\"fs-id1167836518364\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518370\">\n<div data-type=\"problem\" id=\"fs-id1167836518372\">\n<p id=\"fs-id1167836518374\">Why is it important to use the order of operations to simplify an expression?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518386\">\n<div data-type=\"problem\" id=\"fs-id1167836518388\">\n<p id=\"fs-id1167836518390\">Explain how you identify the like terms in the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c258f4bfbd2de54b53d4536a7d74bd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#97;&#43;&#57;&#45;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"170\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518428\">\n<p id=\"fs-id1167836518430\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518436\">\n<div data-type=\"problem\" id=\"fs-id1167836518438\">\n<p id=\"fs-id1167836518440\">Explain the difference between the phrases \u201c4 times the sum of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>\u201d and \u201cthe sum of 4 times <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>\u201d.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836524988\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167836524994\"><span class=\"token\">\u24d0<\/span> Use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836525008\" data-alt=\"This table has 4 columns, 7 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: identify multiples and apply divisibility tests, find prime factorizations and least common multiples, use variables and algebraic symbols, simplify expressions using the order of operations, evaluate an expression, identify and combine like terms, translate English phrases to algebraic expressions. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_01_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 7 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: identify multiples and apply divisibility tests, find prime factorizations and least common multiples, use variables and algebraic symbols, simplify expressions using the order of operations, evaluate an expression, identify and combine like terms, translate English phrases to algebraic expressions. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167836525017\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p>\n<p id=\"fs-id1167836525024\"><strong data-effect=\"bold\">\u2026confidently.<\/strong> Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p>\n<p id=\"fs-id1167836525034\"><strong data-effect=\"bold\">\u2026with some help.<\/strong> This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p>\n<p id=\"fs-id1167836525045\"><strong data-effect=\"bold\">\u2026no &#8211; I don\u2019t get it!<\/strong> This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167836525060\">\n<dt>coefficient<\/dt>\n<dd id=\"fs-id1167836525065\">The coefficient of a term is the constant that multiplies the variable in a term.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836525069\">\n<dt>composite number<\/dt>\n<dd id=\"fs-id1167836525075\">A composite number is a counting number that is not prime. It has factors other than 1 and the number itself.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836525080\">\n<dt>constant<\/dt>\n<dd id=\"fs-id1167836525085\">A constant is a number whose value always stays the same.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836525090\">\n<dt>divisible by a number<\/dt>\n<dd id=\"fs-id1167836525095\">If a number <em data-effect=\"italics\">m<\/em> is a multiple of <em data-effect=\"italics\">n<\/em>, then <em data-effect=\"italics\">m<\/em> is divisible by <em data-effect=\"italics\">n<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836525120\">\n<dt>equation<\/dt>\n<dd id=\"fs-id1167836525125\">An equation is two expressions connected by an equal sign.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836525130\">\n<dt>evaluate an expression<\/dt>\n<dd id=\"fs-id1167836525135\">To evaluate an expression means to find the value of the expression when the variables are replaced by a given number.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836525140\">\n<dt>expression<\/dt>\n<dd id=\"fs-id1167836525146\">An expression is a number, a variable, or a combination of numbers and variables using operation symbols.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836525151\">\n<dt>factors<\/dt>\n<dd id=\"fs-id1167836525156\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4752e3c7f6f79fe949cc961f5a008188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are factors of <em data-effect=\"italics\">m<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018659\">\n<dt>least common multiple<\/dt>\n<dd id=\"fs-id1167833018664\">The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018669\">\n<dt>like terms<\/dt>\n<dd id=\"fs-id1167833018675\">Terms that are either constants or have the same variables raised to the same powers are called like terms.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018680\">\n<dt>multiple of a number<\/dt>\n<dd id=\"fs-id1167833018685\">A number is a multiple of <em data-effect=\"italics\">n<\/em> if it is the product of a counting number and <em data-effect=\"italics\">n.<\/em><\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018699\">\n<dt>order of operations<\/dt>\n<dd id=\"fs-id1167833018704\">The order of operations are established guidelines for simplifying an expression.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018708\">\n<dt>prime factorization<\/dt>\n<dd id=\"fs-id1167833018714\">The prime factorization of a number is the product of prime numbers that equals the number.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018719\">\n<dt>prime number<\/dt>\n<dd id=\"fs-id1167833018724\">A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018730\">\n<dt>simplify an expression<\/dt>\n<dd id=\"fs-id1167833018735\">To simplify an expression means to do all the math possible.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018739\">\n<dt>term<\/dt>\n<dd id=\"fs-id1167833018744\">A term is a constant, or the product of a constant and one or more variables.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833018749\">\n<dt>variable<\/dt>\n<dd id=\"fs-id1167833018754\">A variable is a letter that represents a number whose value may change.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-818","chapter","type-chapter","status-publish","hentry"],"part":762,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/818","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/818\/revisions"}],"predecessor-version":[{"id":15246,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/818\/revisions\/15246"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/762"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/818\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=818"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=818"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=818"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=818"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}