The ability to work comfortably with negative numbers is essential to success in algebra. For this reason, a quick review of adding, subtracting, multiplying, and dividing integers is necessary.Read about The History of Integers. IntegersThe word "integer" is derived from the Latin word integer, which means "whole." Integers are written without using a fractional component. Examples are 2, 3, 1042, 28, 0, −42, −2. Numbers that are fractional—such as , 0.33, and 1.42—are not integers. are all the positive whole numbers, all the negative whole numbers, and zero. As this is intended to be a review of integers, descriptions and examples will not be as detailed as in a normal lesson.

When adding integers, there are two cases to consider. The first is when the signs match—that is, the two integers are both positive or both negative.

If the signs match, add the numbers together and retain the sign.

Example 1.1.2

Add

   

The second case is when the signs don’t match, and there is one positive and one negative number. Subtract the numbers (as if they were all positive), then use the sign from the number with the greatest absolute value. This means that, if the number with the greater absolute value is positive, the answer is positive. If it is negative, the answer is negative.

Example 1.1.5

Add

Example 1.1.6

Add

For subtraction of negatives, change the problem to an addition problem, which is then solved using the above methods. The way to change a subtraction problem to an addition problem is by adding the opposite of the number after the subtraction sign to the number before the subtraction sign. Often, this method is referred to as “adding the opposite.”

Example 1.1.7

Subtract

Example 1.1.8

Subtract

Example 1.1.9

Subtract

Example 1.1.10

Subtract

Multiplication and division of integers both work in a very similar pattern. The short description of the process is to multiply and divide like normal. If the signs match (numbers are both positive or both negative), the answer is positive. If the signs don’t match (one positive and one negative), then the answer is negative.

Example 1.1.11

Multiply

Example 1.1.12

Divide

Questions

For questions 1 to 30, find the sum and/or difference.

For questions 31 to 44, find each product.

For questions 45 to 58, find each quotient.

Answer Key 1.1

Working with fractions is a very important foundational skill in algebra. This section will briefly review reducing, multiplying, dividing, adding, and subtracting fractions. As this is a review, concepts will not be explained in as much detail as they are in other lessons. Final answers of questions working with fractions tend to always be reduced. Reducing fractions is simply done by dividing both the numerator and denominator by the same number.

The previous example could have been done in one step by dividing both the numerator and the denominator by 12. Another solution could have been to divide by 2 twice and then by 3 once (in any order). It is not important which method is used as long as the fraction is reduced as much as possible.

The easiest operation to complete with fractions is multiplication. Fractions can be multiplied straight across, meaning all numerators and all denominators are multiplied together.

Before multiplying, fractions can be reduced. It is possible to reduce vertically within a single fraction, or diagonally within several fractions, as long as one number from the numerator and one number from the denominator are used.

Example 1.2.3

Multiply

Dividing fractions is very similar to multiplying, with one extra step. Dividing fractions necessitates first taking the reciprocal of the second fraction. Once this is done, multiply the fractions together. This multiplication problem solves just like the previous problem.

To add and subtract fractions, it is necessary to first find the least common denominator (LCD). There are several ways to find the LCD. One way is to break the denominators into primes, write out the primes that make up the first denominator, and only add primes that are needed to make the other denominators.

Adding and subtracting fractions is identical in process. If both fractions already have a common denominator, simply add or subtract the numerators and keep the denominator.

While can be written as the mixed number , algebra almost never uses mixed numbers. For this reason, always use the improper fraction, not the mixed number.

If the denominators do not match, it is necessary to first identify the LCD and build up each fraction by multiplying the numerator and denominator by the same number so each denominator is built up to the LCD.

Example 1.2.8

Add

Questions

For questions 1 to 18, simplify each fraction. Leave your answer as an improper fraction.

For questions 19 to 36, find each product. Leave your answer as an improper fraction.

For questions 37 to 52, find each quotient. Leave your answer as an improper fraction.

For questions 53 to 70, evaluate each expression. Leave your answer as an improper fraction.

Answer Key 1.2

When simplifying expressions, it is important to do so in the correct order. Consider the problem 2 + 5 ⋅ 3 done two different ways:

The previous example illustrates that if the same problem is done two different ways, it will result in two different solutions. However, only one method can be correct. It turns out the second method is the correct one. The order of operations ends with the most basic of operations, addition (or subtraction). Before addition is completed, do all repeated addition, also known as multiplication (or division). Before multiplication is completed, do all repeated multiplication, also known as exponents. When something is supposed to be done out of order, to make it come first, put it in parentheses (or grouping symbols). This list, then, is the order of operations used to simplify expressions.

Multiplication and division are on the same level because they are the same operation (division is just multiplying by the reciprocal). This means multiplication and division must be performed from left to right. Therefore, division will come first in some problems, and multiplication will come first in others. The same is true for adding and subtracting (subtracting is just adding the opposite).

Often, students use the word BEMDAS to remember the order of operations, as the first letter of each operation creates the word (written as B E MD AS). Remember BEMDAS to ensure that multiplication and division are done from left to right (same with addition and subtraction).

It is very important to remember to multiply and divide from left to right!

If there are several sets of parentheses in a problem, start with the innermost set and work outward. Inside each set of parentheses, simplify using the order of operations. To make it easier to know which left parenthesis goes with which right parenthesis, different types of grouping symbols will be used, such as braces { }, brackets [ ], and parentheses ( ). These all do the same thing: they are grouping symbols and must be evaluated first.

As Example 1.3.3 illustrates, it can take several steps to complete a problem. The key to successfully solving order of operations problems is to take the time to show your work and do one step at a time. This will reduce the chance of making a mistake along the way.

There are several types of grouping symbols that can be used besides parentheses, brackets, and braces. One such symbol is a fraction bar. The entire numerator and the entire denominator of a fraction must be evaluated before reducing. Once the fraction is reduced, the numerator and denominator can be simplified at the same time.

Another type of grouping symbol is the absolute value. Everything inside a set of absolute value brackets must be evaluated, just as if it were a normal set of parentheses. Then, once the inside is completed, take the absolute value—or distance from zero—to make the number positive.

Questions

For questions 1 to 24, reduce and solve the following expressions.

Answer Key 1.3

When doing algebra, it is common not to know the value of the variables. In this case, simplify where possible and leave any unknown variables in the final solution. One way to simplify expressions is to combine like terms.

Like terms are terms whose variables match exactly, exponents included. Examples of like terms would be and and or −3 and 5. To combine like terms, add (or subtract) the numbers in front of the variables and keep the variables the same.

When combining like terms, subtraction signs must be interpreted as part of the terms they precede. This means that the term following a subtraction sign should be treated like a negative term. The sign always stays with the term.

Another method to simplify is known as distributing. Sometimes, when working with problems, there will be a set of parentheses that makes solving a problem difficult, if not impossible. To get rid of these unwanted parentheses, use the distributive property and multiply the number in front of the parentheses by each term inside.

Several examples of using the distributive property are given below.

In the previous example, it is necessary to again use the fact that the sign goes with the number. This means −6 is treated as a negative number, which gives (−7)(−6) = 42, a positive number. The most common error in distributing is a sign error. Be very careful with signs! It is possible to distribute just a negative throughout parentheses. If there is a negative sign in front of parentheses, think of it like a −1 in front and distribute it throughout.

Distributing throughout parentheses and combining like terms can be combined into one problem. Order of operations says to multiply (distribute) first, then add or subtract (combine like terms). Thus, do each problem in two steps: distribute, then combine.

In Example 1.4.6, −2 is distributed, not just 2. This is because a number being subtracted must always be treated like it has a negative sign attached to it. This makes a big difference, for in that example, when the −5 inside the parentheses is multiplied by −2, the result is a positive number. More involved examples of distributing and combining like terms follow.

Questions

For questions 1 to 28, reduce and combine like terms.

For questions 29 to 58, simplify each expression.

Answer Key 1.4

Digits can be defined as the alphabet of the Hindu–Arabic numeral system that is in common usage today. This alphabet is: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Written in set-builder notation, digits are expressed as:

Natural numbers are often called counting numbers and are usually denoted by These numbers start at 1 and carry on to infinity, which is denoted by the symbol ∞. Writing the set of natural numbers in set-builder notation gives:

Whole numbers include the set of natural numbers and zero. Whole numbers are generally designated by In set-builder notation, the set of whole numbers is denoted by:

Integers include the set of all whole numbers and their negatives. This means the set of integers is composed of positive whole numbers, negative whole numbers, and zero (fractions and decimals are not integers). Common symbols used to represent integers are and For this textbook, the symbol will be used to represent integers.

Rational numbers include all integers and all fractions, terminating decimals, and repeating decimals. Every rational number can be written as a fraction where and are integers. Rational numbers are denoted by the symbol In set-builder notation, the set of rational numbers can be informally written as:

Irrational numbers include any number that cannot be defined by the fraction where and are integers. These are numbers that are non-repeating or non-terminating. Classic examples of irrational numbers are pi  and the square roots of 2 and 3. The symbol for irrational numbers is commonly given as or For this textbook, the symbol will be used. In set-builder notation, the set of irrational numbers can be informally written as:

Real numbers include the set of all rational numbers and irrational numbers. The symbol for real numbers is commonly given as In set-builder notation, the set of real numbers can be informally written as:

Numbers that may not yet have been encountered are imaginary numbers (commonly sometimes ) and complex numbers These numbers will be properly defined later in the textbook.

Imaginary numbers include any real number multiplied by the square root of −1.

Complex numbers are combinations of any real number, imaginary number, or a sum and difference of them.

Consecutive integers are integers that follow each other sequentially. Examples are:

Consecutive even or odd integers are numbers that skip the odd/even sequence to just show odd, odd, odd, or even, even, even. Examples are:

Prime numbers are numbers that cannot be divided by any integer other than 1 and itself. The following is a list of all the prime numbers that are found between 0 and 1000. (Note: 1 is not considered prime.)

Squares are numbers multiplied by themselves. A number that is being squared is shown as having a superscript 2 attached to it. For example, 5 squared is written as 5², which equals 5 × 5 or 25.

Perfect squares are squares of whole numbers, such as 1, 4, 9, 16, 25, 36, and 49. They are found by squaring natural numbers. The following is the list of perfect squares using numbers up to 20:

Cubes are numbers multiplied by themselves three times. A number that is being cubed is shown as having a superscript 3 attached to it. For example, 5 cubed is written as 5³, which equals 5 × 5 × 5 or 125.

Perfect cubes are cubes of whole numbers, such as 1, 8, 27, 64, 125, 216, and 343. They are found by cubing natural numbers. The following is the list of perfect cubes using numbers up to 20:

Percentage means parts per hundred. A percentage can be thought of as a fraction where the numerator, is the number to the left of the % sign, and the denominator, is 100. For example:

Absolute values. The absolute value of an expression denoted is the distance from zero of the number or operation that occurs between the absolute value signs. For example:

Examples of absolute values of simple operations are:

Set-builder notation follows standard patterns and is as follows:

• Begin the set with a left brace {
• A vertical bar | means “such that”
• End the set with a right brace }

So to say is an integer, write this as:

This means “the set of such that is an integer.”

Another way of writing this is to use the symbols that mean “element of” and “not an element of.”

“Element of” is shown by the symbol ∈, and “not an element of” is shown by the element symbol with a line drawn through it, ∉.

In simplest terms, if something is an element of something else, it means that it belongs to or is part of it. For example, a set of numbers called can only be made up of any natural number like 4, 6, 9, and 15. This can be stated as which reads as “the set of such that is an element of the natural number system.”

“Not an element of” can be used to state that the set cannot contain excluded values. For example, say there is a set of all numbers except counting numbers This can be written as:

This can be read as “the set of such that is an element of the set of all real numbers, excluding those numbers that are natural numbers.”

Sets of numbers giving excluded values can be seen throughout this textbook. The standard example is to exclude values that would result in a denominator of zero. This exclusion avoids division by zero and getting an undefinable answer.

The empty set. Sometimes, a set contains no elements. This set is termed the “empty set” or the “null set.” To represent this, write either { } or Ø.

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.

The same process can be used to convert problems with several units in them. Consider the following example.

The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.

Metric and Imperial (U.S.) Conversions

Distance

Imperial to metric conversions:

Area

Imperial to metric conversions:

Volume

Imperial to metric conversions:

Mass

Imperial to metric conversions:

Temperature

Fahrenheit to Celsius conversions:

Celsius to Fahrenheit conversion scale. [Long Description]

Questions

For questions 1 to 18, use dimensional analysis to perform the indicated conversions.

1. 7 miles to yards
2. 234 oz to tons
3. 11.2 mg to grams
4. 1.35 km to centimetres
5. 9,800,000 mm to miles
6. 4.5 ft² to square yards
7. 435,000 m² to square kilometres
8. 8 km² to square feet
9. 0.0065 km³ to cubic metres
10. 14.62 in³ to square centimetres
11. 5500 cm³ to cubic yards
12. 3.5 mph (miles per hour) to feet per second
13. 185 yd per min. to miles per hour
14. 153 ft/s (feet per second) to miles per hour
15. 248 mph to metres per second
16. 186,000 mph to kilometres per year
17. 7.50 tons/yd² to pounds per square inch
18. 16 ft/s² to kilometres per hour squared

For questions 19 to 27, solve each conversion word problem.

19. On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?
20. A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day.
21. An average human heart beats 60 times per minute. If the average person lives to the age of 86, how many times does the average heart beat in a lifetime?
22. Blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. If a person’s blood sugar level measured 128 mg/dL, what is this in grams per litre?
23. You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?
24. A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic metres.
25. A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy more than ¼ of the lot it is built on. Suppose you own a -acre lot (1 acre = 43,560 ft²). What is the maximum allowed footprint for your house in square feet? In square metres?
    
26. A car travels 23 km in 15 minutes. How fast is it going in kilometres per hour? In metres per second?
27. The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams. What is the mass of this diamond in milligrams? Weight in pounds?

Answer Key 1.6

Long Descriptions

Celsius to Fahrenheit conversion scale long description: Scale showing conversions between Celsius and Fahrenheit. The following table summarizes the data:

[Return to Celsius to Fahrenheit conversion scale]

Solving linear equations is an important and fundamental skill in algebra. In algebra, there are often problems in which the answer is known, but the variable part of the problem is missing. To find this missing variable, it is necessary to follow a series of steps that result in the variable equalling some solution.

Addition and Subtraction Problems

To solve equations, the general rule is to do the opposite of the order of operations. Consider the following.

Multiplication Problems

In a multiplication problem, get rid of the coefficient in front of the variable by dividing both sides of the equation by that number. Consider the following examples.

Division Problems

In division problems, remove the denominator by multiplying both sides by it. Consider the following examples.

Questions

For questions 1 to 28, solve each linear equation.

Answer Key 2.1

Extra Reading and Instructional Videos

Article to read: New theory finds ‘traffic jams’ in jet stream cause abnormal weather patterns.

The abstract reads:

A study offers an explanation for a mysterious and sometimes deadly weather pattern in which the jet stream, the global air currents that circle the Earth, stalls out over a region. Much like highways, the jet stream has a capacity, researchers said, and when it’s exceeded, blockages form that are remarkably similar to traffic jams — and climate forecasters can use the same math to model them both.

When working with questions that require two or more steps to solve, do the reverse of the order of operations to solve for the variable.

In solving the above equation, notice that the general pattern followed was to do the opposite of the equation. was added to 16, so 16 was then subtracted from both sides. The variable was multiplied by 4, so both sides were divided by 4.

Questions

For questions 1 to 20, solve each linear equation.

Answer Key 2.2

When working with linear equations with parentheses, the first objective is to isolate the parentheses. Once isolated, the parentheses can be removed and then the variable solved.

For some problems, it is too difficult to isolate the parentheses. In these problems, it is necessary to multiply or divide throughout the parentheses by whatever coefficient is in front of it.

Questions

For questions 1 to 26, solve each linear equation.

Answer Key 2.3

When working with fractions built into linear equations, it is often easiest to remove the fraction in the very first step. This generally means finding the LCD of the fraction and then multiplying every term in the entire equation by the LCD.

Example 2.4.2

Solve for in the equation

First, remove the outside denominator 2 by multiplying both sides by 2:

   

   

Now divide both sides by 3, which leaves:

   

To remove the 9 and 27, multiply both sides by the LCD, 27:

   

This leaves:

   

Questions

For questions 1 to 18, solve each linear equation.

Answer Key 2.4

When solving equations with absolute values, there can be more than one possible answer. This is because the variable whose absolute value is being taken can be either negative or positive, and both possibilities must be accounted for when solving equations.

Example 2.5.1

Solve

   

When there are absolute values in a problem, it is important to first isolate the absolute value, then remove the absolute value by considering both the positive and negative solutions. Notice that, in the next two examples, all the numbers outside of the absolute value are moved to one side first before the absolute value bars are removed and both positive and negative solutions are considered.

Example 2.5.2

Solve

   

Example 2.5.3

Solve

   

Note: the objective in solving for absolute values is to isolate the absolute value to yield a solution.

Example 2.5.4

Solve

   

Often, the absolute value of more than just a variable is being taken. In these cases, it is necessary to solve the resulting equations before considering positive and negative possibilities. This is shown in the next example.

Example 2.5.5

Solve

Since absolute value can be positive or negative, this means that there are two equations to solve.

   

Remember: the absolute value must be isolated first before solving.

Example 2.5.6

Solve

   

Now, solve two equations to get the positive and negative solutions:

   

There exist two other possible results from solving an absolute value besides what has been shown in the above six examples.

Example 2.5.7

Consider the equation from Example 2.5.5.

What happens if, instead, the equation to solve is or

For there is no so there will be just one solution instead of two. Solving this equation yields:

   

For the result will be “no solution” or , since an absolute value is never negative.

One final type of absolute value problem covered in this chapter is when two absolute values are equal to each other. There will still be both a positive and a negative result—the difference is that a negative must be distributed into the second absolute value for the negative possibility.

Example 2.5.8

Solve

Solving this means solving:

   

Removing the absolute value signs leaves:

Questions

For questions 1 to 24, solve each absolute value equation.

Answer Key 2.5

In algebra, expressions often need to be simplified to make them easier to use. There are three basic forms of simplifying, which will be reviewed here. The first form of simplifying expressions is used when the value of each variable in an expression is known. In this case, each variable can be replaced with the equivalent number, and the rest of the expression can be simplified using the order of operations.

Example 2.6.1

Evaluate when and

Whenever a variable is replaced with something, the new number is written inside a set of parentheses. Notice the values of 3 and 5 in the previous example are in parentheses. This is to preserve operations that are sometimes lost in a simple substitution. Sometimes, the parentheses won’t make a difference, but it is a good habit to always use them to prevent problems later.

Example 2.6.2

Evaluate when and

   

Isolating variables in formulas is similar to solving general linear equations. The only difference is, with a formula, there will be several variables in the problem, and the goal is to solve for one specific variable. For example, consider solving a formula such as (the formula for the surface area of a right circular cone) for the variable This means isolating the so the equation has on one side. So a solution might look like This second equation gives the same information as the first; they are algebraically equivalent. However, one is solved for the area while the other is solved for the slant height of the cone

When solving a formula for a variable, focus on the one variable that is being solved for; all the others are treated just like numbers. This is shown in the following example. Two parallel problems are shown: the first is a normal one-step equation, and the second is a formula that you are solving for 

Example 2.6.3

Isolate the variable in the following equations.

   

The same process is used to isolate in as in Because is being solved for, treat all other variables as numbers. For these two equations, both sides were divided by 3 and respectively. A similar idea is seen in the following example.

Example 2.6.4

Isolate the variable in the equation

To isolate the variable must be removed, which is done by subtracting from both sides:

   

Since and are not like terms, they cannot be combined. For this reason, leave the expression as

Example 2.6.5

Isolate the variable in the equation

This means that must be isolated from the variable

   

If no individual term inside parentheses is being solved for, keep the terms inside them together and divide by them as a unit. However, if an individual term inside parentheses is being solved for, it is necessary to distribute. The following example is the same formula as in Example 2.6.5, but this time, is being solved for.

Example 2.6.6

Isolate the variable in the equation

First, distribute throughout :

   

Remove the term from both sides:

   

is then divided by :

   

The solution is which can also be shown as

Be very careful when isolating not to try and cancel the on the top and the bottom of the fraction. This is not allowed if there is any adding or subtracting in the fraction. There is no reducing possible in this problem, so the final reduced answer remains The next example is another two-step problem.

Example 2.6.7

Isolate the variable in the equation

First, subtract from both sides:

   

Now divide both sides by :

   

Therefore, the solution is

It is important to note that a problem is complete when the variable being solved for is isolated or alone on one side of the equation and it does not appear anywhere on the other side of the equation.

The next example is also a two-step equation. It is a problem from earlier in the lesson.

Example 2.6.8

Isolate the variable in the equation

Subtract from both sides:

   

Divide both sides by :

   

The solution is:

   

Formulas often have fractions in them and can be solved in much the same way as any fraction. First, identify the LCD, and then multiply each term by the LCD. After reducing, there will be no more fractions in the problem.

Example 2.6.9

Isolate the variable in the equation

To clear the fraction, multiply both sides by :

   

This leaves:

   

Divide both sides by 2:

   

Which reduces to:

   

Example 2.6.10

Isolate the variable in the equation

To clear the fraction, multiply both sides by :

   

Which reduces to:

   

Distribute throughout then isolate:

   

Finally, divide both sides by :

   

Solution:

   

Questions

For questions 1 to 10, evaluate each expression using the values given.

For questions 11 to 34, isolate the indicated variable from the equation.

Answer Key 2.6

Direct Variation Problems

There are many mathematical relations that occur in life. For instance, a flat commission salaried salesperson earns a percentage of their sales, where the more they sell equates to the wage they earn. An example of this would be an employee whose wage is 5% of the sales they make. This is a direct or a linear variation, which, in an equation, would look like:

A historical example of direct variation can be found in the changing measurement of pi, which has been symbolized using the Greek letter π since the mid 18th century. Variations of historical π calculations are Babylonian Egyptian and Indian In the 5th century, Chinese mathematician Zu Chongzhi calculated the value of π to seven decimal places (3.1415926), representing the most accurate value of π for over 1000 years.

Pi is found by taking any circle and dividing the circumference of the circle by the diameter, which will always give the same value: 3.14159265358979323846264338327950288419716… (42 decimal places). Using an infinite-series exact equation has allowed computers to calculate π to 10¹³ decimals.

All direct variation relationships are verbalized in written problems as a direct variation or as directly proportional and take the form of straight line relationships. Examples of direct variation or directly proportional equations are:

• 
  • varies directly as
  • varies as
  • varies directly proportional to
  • is proportional to
• 
  • varies directly as the square of
  • varies as squared
  • is proportional to the square of
• 
  • varies directly as the cube of
  • varies as cubed
  • is proportional to the cube of
• 
  • varies directly as the square root of
  • varies as the root of
  • is proportional to the square root of

Inverse Variation Problems

Inverse variation problems are reciprocal relationships. In these types of problems, the product of two or more variables is equal to a constant. An example of this comes from the relationship of the pressure and the volume of a gas, called Boyle’s Law (1662). This law is written as:

Written as an inverse variation problem, it can be said that the pressure of an ideal gas varies as the inverse of the volume or varies inversely as the volume. Expressed this way, the equation can be written as:

Another example is the historically famous inverse square laws. Examples of this are the force of gravity electrostatic force and the intensity of light In all of these measures of force and light intensity, as you move away from the source, the intensity or strength decreases as the square of the distance.

In equation form, these look like:

These equations would be verbalized as:

• The force of gravity varies inversely as the square of the distance.
• Electrostatic force varies inversely as the square of the distance.
• The intensity of a light source varies inversely as the square of the distance.

All inverse variation relationship are verbalized in written problems as inverse variations or as inversely proportional. Examples of inverse variation or inversely proportional equations are:

• 
  • varies inversely as
  • varies as the inverse of
  • varies inversely proportional to
  • is inversely proportional to
• 
  • varies inversely as the square of
  • varies inversely as squared
  • is inversely proportional to the square of
• 
  • varies inversely as the cube of
  • varies inversely as cubed
  • is inversely proportional to the cube of
• 
  • varies inversely as the square root of
  • varies as the inverse root of
  • is inversely proportional to the square root of

Joint or Combined Variation Problems

In real life, variation problems are not restricted to single variables. Instead, functions are generally a combination of multiple factors. For instance, the physics equation quantifying the gravitational force of attraction between two bodies is:

where:

• stands for the gravitational force of attraction
• is Newton’s constant, which would be represented by in a standard variation problem
• and are the masses of the two bodies
• is the distance between the centres of both bodies

To write this out as a variation problem, first state that the force of gravitational attraction between two bodies is directly proportional to the product of the two masses and inversely proportional to the square of the distance separating the two masses. From this information, the necessary equation can be derived. All joint variation relationships are verbalized in written problems as a combination of direct and inverse variation relationships, and care must be taken to correctly identify which variables are related in what relationship.

Solving these combined or joint variation problems is the same as solving simpler variation problems.

First, decide what equation the variation represents. Second, break up the data into the first data given—which is used to find —and then the second data, which is used to solve the problem given. Consider the following joint variation problem.

Questions

For questions 1 to 12, write the formula defining the variation, including the constant of variation

1. varies directly as
2. is jointly proportional to and
3. varies inversely as
4. varies directly as the square of
5. varies jointly as and
6. is inversely proportional to the cube of
7. is jointly proportional with the square of and the square root of
8. is inversely proportional to to the sixth power
9. is jointly proportional with the cube of and inversely to the square root of
10. is inversely proportional with the square of and the square root of
11. varies jointly as and and is inversely proportional to the cube of
12. is inversely proportional to the cube of and square of

For questions 13 to 22, find the formula defining the variation and the constant of variation

13. If varies directly as find when and
14. If is jointly proportional to and find when and
15. If varies inversely as find when and
16. If varies directly as the square of find when and
17. If varies jointly as and find when and
18. If is inversely proportional to the cube of find when and
19. If is directly proportional to find when and
20. If is jointly proportional with the square of and the square root of find when and
21. If varies jointly with and the square of and inversely with find when and
22. If varies directly as and inversely as find when and

For questions 23 to 37, solve each variation word problem.

23. The electrical current (in amperes, A) varies directly as the voltage in a simple circuit. If the current is 5 A when the source voltage is 15 V, what is the current when the source voltage is 25 V?
24. The current in an electrical conductor varies inversely as the resistance (in ohms, Ω) of the conductor. If the current is 12 A when the resistance is 240 Ω, what is the current when the resistance is 540 Ω?
25. Hooke’s law states that the distance that a spring is stretched supporting a suspended object varies directly as the mass of the object If the distance stretched is 18 cm when the suspended mass is 3 kg, what is the distance when the suspended mass is 5 kg?
26. The volume of an ideal gas at a constant temperature varies inversely as the pressure exerted on it. If the volume of a gas is 200 cm³ under a pressure of 32 kg/cm², what will be its volume under a pressure of 40 kg/cm²?
27. The number of aluminum cans used each year varies directly as the number of people using the cans. If 250 people use 60,000 cans in one year, how many cans are used each year in a city that has a population of 1,000,000?
28. The time required to do a masonry job varies inversely as the number of bricklayers If it takes 5 hours for 7 bricklayers to build a park wall, how much time should it take 10 bricklayers to complete the same job?
29. The wavelength of a radio signal (λ) varies inversely as its frequency A wave with a frequency of 1200 kilohertz has a length of 250 metres. What is the wavelength of a radio signal having a frequency of 60 kilohertz?
30. The number of kilograms of water in a human body is proportional to the mass of the body If a 96 kg person contains 64 kg of water, how many kilograms of water are in a 60 kg person?
31. The time required to drive a fixed distance varies inversely as the speed If it takes 5 hours at a speed of 80 km/h to drive a fixed distance, what speed is required to do the same trip in 4.2 hours?
32. The volume of a cone varies jointly as its height and the square of its radius If a cone with a height of 8 centimetres and a radius of 2 centimetres has a volume of 33.5 cm³, what is the volume of a cone with a height of 6 centimetres and a radius of 4 centimetres?
33. The centripetal force acting on an object varies as the square of the speed and inversely to the radius of its path. If the centripetal force is 100 N when the object is travelling at 10 m/s in a path or radius of 0.5 m, what is the centripetal force when the object’s speed increases to 25 m/s and the path is now 1.0 m?
34. The maximum load that a cylindrical column with a circular cross section can hold varies directly as the fourth power of the diameter and inversely as the square of the height If an 8.0 m column that is 2.0 m in diameter will support 64 tonnes, how many tonnes can be supported by a column 12.0 m high and 3.0 m in diameter?
35. The volume of gas varies directly as the temperature and inversely as the pressure If the volume is 225 cc when the temperature is 300 K and the pressure is 100 N/cm², what is the volume when the temperature drops to 270 K and the pressure is 150 N/cm²?
36. The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter A wire with a length of 5.0 m and a diameter of 0.25 cm has a resistance of 20 Ω. Find the electrical resistance in a 10.0 m long wire having twice the diameter.
37. The volume of wood in a tree varies directly as the height and the diameter If the volume of a tree is 377 m³ when the height is 30 m and the diameter is 2.0 m, what is the height of a tree having a volume of 225 m³ and a diameter of 1.75 m?

Answer Key 2.7

The centre number of each square is found by using the order of operations on the numbers that surround it. The challenge is to solve for the variable

Can you solve for ? Can you find any other possible solution?

Answer Key 2.8

Often, to get an idea of the behaviour of an equation or some function, a visual representation that displays the solutions to the equation or function in the form of a graph will be made. Before exploring this, it is necessary to review the foundations of a graph. The following is an example of what is called the coordinate plane of a graph.

The plane is divided into four sections by a horizontal number line (-axis) and a vertical number line (-axis). Where the two lines meet in the centre is called the origin. This centre origin is where and and is represented by the ordered pair .

-axis

For the -axis, moving to the right from the centre 0, the numbers count up, and To the left of the centre 0, the numbers count down, and

-axis

Similarly, for the -axis, moving up from the centre 0, the numbers count up, and Moving down from the centre 0, the numbers count down, and

When identifying points on a graph, a dot is generally used with a set of parentheses following that gives the -value followed by the -value. This will look like or and is given the formal name of an ordered pair.

This coordinate system is universally used, with the simplest example being the kind of treasure map that is usually encountered in childhood, or the longitude and latitude system used to identify any position on the Earth. For this system, the -axis (which represents latitude) is the equator and the -axis (which represents longitude) or the prime meridian is the line that passes though Greenwich, England. The origin of the Earth’s latitude and longitude (0°, 0°) is a fictional island called “Null Island.”

Latitude and longitude. [Long Description]

The treasure map of Robert Louis Stevenson made popular by his work Treasure Island. From Cordingly, David (1995). Under the Black Flag: The romance and the reality of life among the pirates. Times Warner, 1996.

Questions

1. What are the coordinates of each point on the graph below?
   
2. Plot and label the following points on the graph.
   
   

Answer Key 3.1

Long Descriptions

Latitude and longitude long description: Two views of the globe that show the landmark points of the latitude and longitude system.

The first globe demonstrates the lines of latitude. The centre line of latitude is called the equator and represents 0° latitude. It wraps around the centre of the Earth from west to east. The globe shows North and South America, and the equator runs through the northern part of South America. The North Pole is at 90° latitude and the South Pole is at −90° latitude. Positive latitude is above the equator, and negative latitude is below it.

The second globe demonstrates the lines of longitude. The centre line of longitude is called the prime meridian and represents 0° longitude. It wraps around the centre of the Earth from north to south. It passes through Greenwich, England, by convention, as well as parts of France, Spain, and western Africa. Positive longitude is east of the prime meridian, and negative longitude is west of it. [Return to Latitude and longitude]

Finding the Distance Between Two Points

The logic used to find the distance between two data points on a graph involves the construction of a right triangle using the two data points and the Pythagorean theorem to find the distance.

To do this for the two data points and , the distance between these two points will be found using and

Using the Pythagorean theorem, this will end up looking like:

or, in expanded form:

On graph paper, this looks like the following. For this illustration, both and are 7 units long, making the distance or .

The square root of 98 is approximately 9.899 units long.

Finding the Midway Between Two Points (Midpoint)

The logic used to find the midpoint between two data points and on a graph involves finding the average values of the data points and the of the data points . The averages are found by adding both data points together and dividing them by .

In an equation, this looks like:

and

Questions

For questions 1 to 8, find the distance between the points.

1.  (−6, −1) and (6, 4)
2. (1, −4) and (5, −1)
3. (−5, −1) and (3, 5)
4. (6, −4) and (12, 4)
5. (−8, −2) and (4, 3)
6. (3, −2) and (7, 1)
7. (−10, −6) and (−2, 0)
8. (8, −2) and (14, 6)

For questions 9 to 16, find the midpoint between the points.

9. (−6, −1) and (6, 5)
10. (1, −4) and (5, −2)
11. (−5, −1) and (3, 5)
12. (6, −4) and (12, 4)
13. (−8, −1) and (6, 7)
14. (1, −6) and (3, −2)
15. (−7, −1) and (3, 9)
16. (2, −2) and (12, 4)

Answer Key 3.2

Another important property of any line or linear function is slope. Slope is a measure of steepness and indicates in some situations how fast something is changing—specifically, its rate of change. A line with a large slope, such as 10, is very steep. A line with a small slope, such as is very flat or nearly level. Lines that rise from left to right are called positive slopes and lines that sink are called negative slopes. Slope can also be used to describe the direction of a line. A line that goes up as it moves from from left to right is described as having a positive slope whereas a line that goes downward has a negative slope. Slope, therefore, will define a line as rising or falling.

Slopes in real life have significance. For instance, roads with slopes that are potentially dangerous often carry warning signs. For steep slopes that are rising, extra slow moving lanes are generally provided for large trucks. For roads that have steep down slopes, runaway lanes are often provided for vehicles that lose their ability to brake.

When quantifying slope, use the measure of the rise of the line divided by its run. The symbol that represents slope is the letter which has unknown origins. Its first recorded usage is in an 1844 text by Matthew O’Brian, “A Treatise on Plane Co-Ordinate Geometry,”Derivation of Slope: https://services.math.duke.edu//education/webfeats/Slope/Slopederiv.html which was quickly followed by George Salmon’s “A Treatise on Conic Sections” (1848), in which he used in the equation

Since the rise of a line is shown by the change in the -value and the run is shown by the change in the -value, this equation is shortened to:

This equation is often expanded to:

Example 3.3.1

Find the slope of the following line.

First, choose two points on the line on this graph. Any points can be chosen, but they should fall on one of the corner grids. These are labelled and

To find the slope of this line, consider the rise, or vertical change, and the run, or horizontal change. Observe in this example that the -value (the rise) goes from 4 to −2.

Therefore, , or  (4 − −2), which equals (4 + 2), or 6.

The -value (the run) goes from  −2 to 4.

Therefore, , or  (−2 − 4), which equals (−2 + −4), or −6.

This means the slope of this line is , or , or −1.

   

Example 3.3.2

Find the slope of the following line.

First, choose two points on the line on this graph. Any points can be chosen, but to fall on a corner grid, they should be on opposite sides of the graph. These are and

To find the slope of this line, consider the rise, or vertical change, and the run, or horizontal change. Observe in this example that the -value (the rise) goes from −4 to 1.

Therefore, , or  (1 − −4), which equals 5.

The -value (the run) goes from −6 to 6.

Therefore, or (6 − −6), which equals 12.

This means the slope of this line is , or , which cannot be further simplified.

   

There are two special lines that have unique slopes that must be noted: lines with slopes equal to zero and slopes that are undefined.

Undefined slopes arise when the line on the graph is vertical, going straight up and down. In this case, which means that zero is divided by while calculating the slope, which makes it undefined.

Zero slopes are flat, horizontal lines that do not rise or fall; therefore, In this case, the slope is simply 0.

Most often, the slope of the line must be found using data points rather than graphs. In this case, two data points are generally given, and the slope is found by dividing by This is usually done using the expanded slope equation of:

Questions

For questions 1 to 6, find the slope of each line shown on the graph.

For questions 7 to 26, find the slope of the line that would connect each pair of points.

7. (2, 10), (−2, 15)
8. (1, 2), (−6, −12)
9. (−5, 10), (0, 0)
10. (2, −2), (7, 8)
11. (4, 6), (−8, −10)
12. (−3, 6), (9, −6)
13. (−2 −4), (10, −4)
14. (3, 5), (2, 0)
15. (−4, 4), (−6, 8)
16. (9, −6), (−7, −7)
17. (2, −9), (6, 4)
18. (−6, 2), (5, 0)
19. (−5, 0), (−5, 0)
20. (8, 11), (−3, −13)
21. (−7, 9), (1, −7)
22. (1, −2), (1, 7)
23. (7, −4), (−8, −9)
24. (−8, −5), (4, −3)
25. (−5, 7), (−8, 4)
26. (9, 5), (5, 1)

Answer Key 3.3

There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation.

If the equation is given in the form , then gives the rise over run value and the value gives the point where the line crosses the -axis, also known as the -intercept.

When graphing a linear equation using the slope-intercept method, start by using the value given for the -intercept. After this point is marked, then identify other points using the slope.

This is shown in the following example.

The second method of drawing lines represented by linear equations and functions is to identify the two intercepts of the linear equation. Specifically, find when and find when .

Questions

For questions 1 to 10, sketch each linear equation using the slope-intercept method.

For questions 11 to 20, sketch each linear equation using the and -intercepts.

For questions 21 to 28, sketch each linear equation using any method.

For questions 29 to 40, reduce and sketch each linear equation using any method.

Answer Key 3.4

Quite often, students are required to find the equation of a line given only a data point and a slope or two data points. The simpler of these problems is to find the equation when given a slope and a data point. To do this, use the equation that defines the slope of a line:

This becomes:

To illustrate this method, consider the following example.

The more difficult variant of this type of problem is that in which the equation of a line that connects two data points must be found. However, this is simpler than it may seem.

The first step is to find the slope of the line that would connect those two points. Use the slope equation, as has been done previously in this textbook. After this is done, use this slope and one of the two data points given at the beginning of the problem. The following example illustrates this.

Questions

For questions 1 to 12, write the slope-intercept form of each linear equation using the given point and slope.

1. and
2. and
3. and
4. and
5. and
6. and
7. and
8. and
9. and
10. and
11. and
12. and

For questions 13 to 24, write the general form of each linear equation using the given point and slope.

13. and
14. and
15. and
16. and
17. and
18. and
19. and
20. and
21. and
22. and
23. and
24. and

For questions 25 and 32, write the slope-intercept form of each linear equation using the given points.

25. and
26. and
27. and
28. and
29. and
30. and
31. and
32. and

For questions 33 to 40, write the general form of each linear equation using the given points.

33. and
34. and
35. and
36. and
37. and
38. and
39. and
40. and

Answer Key 3.5

Perpendicular, parallel, horizontal, and vertical lines are special lines that have properties unique to each type. Parallel lines, for instance, have the same slope, whereas perpendicular lines are the opposite and have negative reciprocal slopes. Vertical lines have a constant -value, and horizontal lines have a constant -value.

Two equations govern perpendicular and parallel lines:

For parallel lines, the slope of the first line is the same as the slope for the second line. If the slopes of these two lines are called and , then .

Perpendicular lines are slightly more difficult to understand. If one line is rising, then the other must be falling, so both lines have slopes going in opposite directions. Thus, the slopes will always be negative to one another. The other feature is that the slope at which one is rising or falling will be exactly flipped for the other one. This means that the slopes will always be negative reciprocals to each other. If the slopes of these two lines are called and , then .

Typically, questions that are asked of students in this topic are written in the form of “Find the equation of a line passing through point that is perpendicular/parallel to .” The first step is to identify the slope that is to be used to solve this equation, and the second is to use the described methods to arrive at the solution like previously done. For instance:

Questions that are looking for the vertical or horizontal line through a given point are the easiest to do and the most commonly confused.

Vertical lines always have a single -value, yielding an equation like

Horizontal lines always have a single -value, yielding an equation like

Questions

For questions 1 to 6, find the slope of any line that would be parallel to each given line.

For questions 7 to 12, find the slope of any line that would be perpendicular to each given line.

For questions 13 to 18, write the slope-intercept form of the equation of each line using the given point and line.

13. (1, 4) and parallel to
14. (5, 2) and perpendicular to
15. (3, 4) and parallel to
16. (1, −1) and perpendicular to
17. (2, 3) and parallel to
18. (−1, 3) and perpendicular to

For questions 19 to 24, write the general form of the equation of each line using the given point and line.

19. (1, −5) and parallel to
20. (1, −2) and perpendicular to
21. (5, 2) and parallel to
22. (1, 3) and perpendicular to
23. (4, 2) and parallel to
24. (3, −5) and perpendicular to

For questions 25 to 36, write the equation of either the horizontal or the vertical line that runs through each point.

25. Horizontal line through (4, −3)
26. Vertical line through (−5, 2)
27. Vertical line through (−3,1)
28. Horizontal line through (−4, 0)
29. Horizontal line through (−4, −1)
30. Vertical line through (2, 3)
31. Vertical line through (−2, −1)
32. Horizontal line through (−5, −4)
33. Horizontal line through (4, 3)
34. Vertical line through (−3, −5)
35. Vertical line through (5, 2)
36. Horizontal line through (5, −1)

Answer Key 3.6

Number-based word problems can be very confusing, and it takes practice to convert a word-based sentence into a mathematical equation. The best strategy to solve these problems is to identify keywords that can be pulled out of a sentence and use them to set up an algebraic equation.

Variables that are to be solved for are often written as “a number,” “an unknown,” or “a value.”

“Equal” is generally represented by the words “is,” “was,” “will be,” or “are.”

Addition is often stated as “more than,” “the sum of,” “added to,” “increased by,” “plus,” “all,” or “total.” Addition statements are quite often written backwards. An example of this is “three more than an unknown number,” which is written as

Subtraction is often written as “less than,” “minus,” “decreased by,” “reduced by,” “subtracted from,” or “the difference of.” Subtraction statements are quite often written backwards. An example of this is “three less than an unknown number,” which is written as

Multiplication can be seen in written problems with the words “times,” “the product of,” or “multiplied by.”

Division is generally found by a statement such as “divided by,” “the quotient of,” or “per.”

Another type of number problem involves consecutive integers, consecutive odd integers, or consecutive even integers. Consecutive integers are numbers that come one after the other, such as 3, 4, 5, 6, 7. The equation that relates consecutive integers is:

Consecutive odd integers and consecutive even integers both share the same equation, since every second number must be skipped to remain either odd (such as 3, 5, 7, 9) or even (2, 4, 6, 8). The equation that is used to represent consecutive odd or even integers is:

Questions

For questions 1 to 8, write the formula defining each relationship. Do not solve.

1. Five more than twice an unknown number is 25.
2. Twelve more than 4 times an unknown number is 36.
3. Three times an unknown number decreased by 8 is 22.
4. Six times an unknown number less 8 is 22.
5. When an unknown number is decreased by 8, the difference is half the unknown number.
6. When an unknown number is decreased by 4, the difference is half the unknown number.
7. The sum of three consecutive integers is 21.
8. The sum of the first two of three odd consecutive integers, less the third, is 5.

For questions 9 to 16, write and solve the equation describing each relationship.

9. When five is added to three times a certain number, the result is 17. What is the number?
10. If five is subtracted from three times a certain number, the result is 10. What is the number?
11. Sixty more than nine times a number is the same as two less than ten times the number. What is the number?
12. Eleven less than seven times a number is five more than six times the number. Find the number.
13. The sum of three consecutive integers is 108. What are the integers?
14. The sum of three consecutive integers is −126. What are the integers?
15. Find three consecutive integers such that the sum of the first, twice the second, and three times the third is −76.
16. Find three consecutive odd integers such that the sum of the first, two times the second, and three times the third is 70.

Answer Key 3.7

A newspaper delivery driver wants to visit of number of specific stores in a city to drop off bundles of newspapers and return to the starting position by taking the shortest distance. Since there are a number of stores and choices of what path is best to take, what is the shortest distance you can find?

One possible path is shown below. Can you find the distance covered in this path if each square represents 1 km?

Can you find a shorter path? How many km of driving would you save?

These problems can rapidly become more difficult as the number of stops are increased. An example of this is as follows. What is the shortest distance that you can find for this puzzle?

Why UPS Drivers Don’t Turn Left and You Probably Shouldn’t Either

The company UPS has analyzed delivery routs thoroughly. Read this article by Graham KendallKendall, Graham. Why UPS Drivers Don't Turn Right and Why You Probably Shouldn't Either. https://theconversation.com/why-ups-drivers-dont-turn-left-and-you-probably-shouldnt-either-71432 to find out how this affects the routes their drivers take.

Why UPS drivers don’t turn left and you probably shouldn’t either

When given an equation, such as or there are specific values for the variable. However, with inequalities, there is a range of values for the variable rather than a defined value. To write the inequality, use the following notation and symbols:

Symbol	Meaning
	> Greater than
	≤ Greater than or equal to
	< Less than
	≥ Less than or equal to

When solving inequalities, the direction of the inequality sign (called the sense) can flip over. The sense will flip under two conditions:

First, the sense flips when the inequality is divided or multiplied by a negative. For instance, in reducing , it is necessary to divide both sides by −3. This leaves -4." title="Rendered by QuickLaTeX.com" height="13" width="61" style="vertical-align: -1px;">

Second, the sense will flip over if the entire equation is flipped over. For instance, 2" title="Rendered by QuickLaTeX.com" height="12" width="42" style="vertical-align: 0px;">, when flipped over, would look like In both cases, the 2 must be shown to be smaller than the , or the is always greater than 2, no matter which side each term is on.

Inequalities can get as complex as the linear equations previously solved in this textbook. All the same patterns for solving inequalities are used for solving linear equations.

Questions

For questions 1 to 6, draw a graph for each inequality and give its interval notation.

1. -5" title="Rendered by QuickLaTeX.com" height="13" width="56" style="vertical-align: 0px;">
2. 4" title="Rendered by QuickLaTeX.com" height="13" width="43" style="vertical-align: -1px;">
3. 
4. 
5. 
6. 

For questions 7 to 12, write the inequality represented on each number line and give its interval notation.

For questions 13 to 38, draw a graph for each inequality and give its interval notation.

13. 
14. 
15. 
16. 
17. 
18. 8+\dfrac{x}{2}" title="Rendered by QuickLaTeX.com" height="32" width="84" style="vertical-align: -12px;">
19. \dfrac{(a-2)}{5}" title="Rendered by QuickLaTeX.com" height="38" width="88" style="vertical-align: -12px;">
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. -8(x + 2) + 4x" title="Rendered by QuickLaTeX.com" height="18" width="213" style="vertical-align: -4px;">
35. 
36. 
37. -k - 20" title="Rendered by QuickLaTeX.com" height="18" width="154" style="vertical-align: -4px;">
38. 

Answer Key 4.1

Several inequalities can be combined together to form what are called compound inequalities.

The first type of compound inequality is the “or” inequality, which is true when either inequality results in a true statement. When graphing this type of inequality, one useful trick is to graph each individual inequality above the number line before moving them both down together onto the actual number line.

When giving interval notation for a solution, if there are two different parts to the graph, put a ∪ (union) symbol between two sets of interval notation, one for each part.

Note: there are several possible results that result from an “or” statement. The graphs could be pointing different directions, as in the graph above, or pointing in the same direction, as in the graph representing -1 \text{ or }x > 3" title="Rendered by QuickLaTeX.com" height="13" width="127" style="vertical-align: -1px;"> that is shown below.

−1 or x > 3. Left parenthesis at −1; right arrow to infinity." class="aligncenter wp-image-2609" width="431" height="207" srcset="https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_2-300x144.jpg 300w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_2-65x31.jpg 65w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_2-225x108.jpg 225w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_2-350x168.jpg 350w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_2.jpg 539w" sizes="(max-width: 431px) 100vw, 431px">

In interval notation, this solution is written as .

It is also possible to have solutions that point in opposite directions but are overlapping, as shown by the solutions and graph below.

−1 or x < 3. Left and right arrow stretching into negative and positive infinity." class="aligncenter wp-image-2611" width="378" height="150" srcset="https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_3-300x119.jpg 300w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_3-65x26.jpg 65w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_3-225x89.jpg 225w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_3-350x139.jpg 350w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_3.jpg 552w" sizes="(max-width: 378px) 100vw, 378px">

In interval notation, this solution is written as , or simply ,  since the graph is all possible numbers.

The second type of compound inequality is the “and” inequality. “And” inequalities require both inequality statements to be true. If one part is false, the whole inequality is false. When graphing these inequalities, follow a similar process as before, sketching both solutions for both inequalities above the number line. However, this time, it is only the overlapping portion that is drawn onto the number line. When a solution for an “and” compound inequality is given in interval notation, it will be expressed in a manner very similar to single inequalities. The symbol that can be used for “and” is the intersection symbol, ∩.

Note: there are several different results that could result from an “and” statement. The graphs could be pointing towards each other as in the graph above, or pointing in the same direction, as in the graph representing -1 \text{ and }x > 3" title="Rendered by QuickLaTeX.com" height="14" width="140" style="vertical-align: -1px;"> (shown below). In this case, the solution must be true for both inequalities, which make a combined graph of:

−1 and x > 3. Left parenthesis at 3 and right arrow to infinity." class="aligncenter wp-image-2615" width="367" height="181" srcset="https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_5-300x148.jpg 300w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_5-65x32.jpg 65w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_5-225x111.jpg 225w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_5-350x172.jpg 350w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_5.jpg 546w" sizes="(max-width: 367px) 100vw, 367px">

In interval notation, this solution is written as

It is also possible to have solutions that point in opposite directions but do not overlap, as shown by the solutions and graph below. Since there is no overlap, there is no real solution.

1 and x < −3. No overlap, so no real solution." class="aligncenter wp-image-2618" width="436" height="151" srcset="https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_6-300x104.jpg 300w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_6-65x23.jpg 65w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_6-225x78.jpg 225w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_6-350x121.jpg 350w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.2_6.jpg 617w" sizes="(max-width: 436px) 100vw, 436px">

In interval notation, this solution is written as no solution, or .

The third type of compound inequality is a special type of “and” inequality. When the variable (or expression containing the variable) is between two numbers, write it as a single math sentence with three parts, such as to show is greater than 5 and less than or equal to 8. To stay balanced when solving this type of inequality, because there are three parts to work with, it is necessary to perform the same operation on all three parts. The graph, then, is of the values between the benchmark numbers with appropriate brackets on the ends.

Questions

For questions 1 to 32, solve each compound inequality, graph its solution, and write it in interval notation.

1. 
2. 
3. 
4. 0 \text{ or } r - 5 < -12" title="Rendered by QuickLaTeX.com" height="14" width="179" style="vertical-align: -1px;">
5. 
6. 40" title="Rendered by QuickLaTeX.com" height="15" width="162" style="vertical-align: -2px;">
7. -1 \text{ and } v - 2 < 1" title="Rendered by QuickLaTeX.com" height="32" width="169" style="vertical-align: -12px;">
8. 
9. 
10. 
11. 
12. 4" title="Rendered by QuickLaTeX.com" height="16" width="176" style="vertical-align: -3px;">
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 59 \text{ or } -6r - 3 > 33" title="Rendered by QuickLaTeX.com" height="15" width="228" style="vertical-align: -2px;">
21. 
22. 82" title="Rendered by QuickLaTeX.com" height="15" width="235" style="vertical-align: -3px;">
23. 
24. 
25. 
26. 
27. 
28. 
29. 2k + 10" title="Rendered by QuickLaTeX.com" height="16" width="280" style="vertical-align: -3px;">
30. 
31. 
32. 

Answer Key 4.2

Absolute values are positive magnitudes, which means that they represent the positive value of any number.

For instance, | −5 | and | +5 | are the same, with both having the same value of 5, and | −99 | and | +99 | both share the same value of 99.

When used in inequalities, absolute values become a boundary limit to a number.

When drawing the boundaries for inequalities on a number line graph, use the following conventions:

For ≤ or ≥, use [brackets] as boundary limits.

For < or >, use (parentheses) as boundary limits.

Equation	Number Line
4" title="Rendered by QuickLaTeX.com" height="18" width="51" style="vertical-align: -4px;">	4. Right parenthesis on −4; left parenthesis on 4. Arrows to both infinities." class="alignnone wp-image-2740" width="336" height="56" srcset="https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.3_5-300x50.jpg 300w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.3_5-65x11.jpg 65w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.3_5-225x37.jpg 225w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.3_5-350x58.jpg 350w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.3_5.jpg 489w" sizes="(max-width: 336px) 100vw, 336px">

When an inequality has an absolute value, isolate the absolute value first in order to graph a solution and/or write it in interval notation. The following examples will illustrate isolating and solving an inequality with an absolute value.

Other examples of absolute value inequalities result in an algebraic expression that is bounded by an inequality.

It is important to remember when solving these equations that the absolute value is always positive. If given an absolute value that is less than a negative number, there will be no solution because absolute value will always be positive, i.e., greater than a negative. Similarly, if absolute value is greater than a negative, the answer will be all real numbers.

This means that:

-6 \text{ has every number as a solution and is written as } (-\infty, \infty) \end{array}" title="Rendered by QuickLaTeX.com" height="106" width="563" style="vertical-align: -48px;">

Note: since infinity can never be reached, use parentheses instead of brackets when writing infinity (positive or negative) in interval notation.

Questions

For questions 1 to 33, solve each inequality, graph its solution, and give interval notation.

1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 3" title="Rendered by QuickLaTeX.com" height="18" width="123" style="vertical-align: -4px;">
14. 5" title="Rendered by QuickLaTeX.com" height="18" width="50" style="vertical-align: -4px;">
15. 5" title="Rendered by QuickLaTeX.com" height="18" width="58" style="vertical-align: -4px;">
16. 5" title="Rendered by QuickLaTeX.com" height="18" width="80" style="vertical-align: -4px;">
17. 3" title="Rendered by QuickLaTeX.com" height="18" width="81" style="vertical-align: -4px;">
18. 6" title="Rendered by QuickLaTeX.com" height="18" width="90" style="vertical-align: -4px;">
19. 3" title="Rendered by QuickLaTeX.com" height="18" width="81" style="vertical-align: -4px;">
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. -4" title="Rendered by QuickLaTeX.com" height="18" width="167" style="vertical-align: -4px;">

Answer Key 4.3

Graphing a 2D Inequality

To graph an inequality, borrow the strategy used to draw a line graph in 2D. To do this, replace the inequality with an equal sign.

Example 4.4.1

Consider the following inequalities:

    &12 \\ 3x&+&2y&\ge &12 \end{array}\]" title="Rendered by QuickLaTeX.com">

All can be changed to by replacing the inequality sign with =.

It is then possible to create a data table using the new equation.

Create a data table of values for the equation

0	6
2	3
4	0
6	−3

Using these values, plot the data points on a graph.

Once the data points are plotted, draw a line that connects them all. The type of line drawn depends on the original inequality that was replaced.

If the inequality had ≤ or ≥, then draw a solid line to represent data points that are on the line.

If the inequality had < or >, then draw a dashed line instead to indicate that some data points are excluded.

If the inequality is either or , then draw its graph using a solid line and solid dots.

If the inequality is either or 12" title="Rendered by QuickLaTeX.com" height="16" width="99" style="vertical-align: -4px;">, then draw its graph using a dashed line and hollow dots.

There remains only one step to complete this graph: finding which side of the line makes the inequality true and shading it. The easiest way to do this is to choose the data point .

It is evident that, for and , the data point is true for the inequality. In this case, shade the side of the line that contains the data point .

It is also clear that, for and 12" title="Rendered by QuickLaTeX.com" height="18" width="125" style="vertical-align: -4px;">, the data point is false for the inequality. In this case, shade the side of the line that does not contain the data point .

12. The side without (0, 0) is shaded." width="300" height="292" srcset="https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.4_7jpg-300x292.jpg 300w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.4_7jpg-65x63.jpg 65w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.4_7jpg-225x219.jpg 225w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.4_7jpg-350x340.jpg 350w, https://pressbooks.bccampus.ca/intermediatealgebrakpu/wp-content/uploads/sites/653/2019/04/Chapter4.4_7jpg.jpg 425w" sizes="(max-width: 300px) 100vw, 300px">

Graphing an Absolute Value Function

To graph an absolute value function, first create a data table using the absolute value part of the equation.

The data point that is started with is the one that makes the absolute value equal to 0 (this is the -value of the vertex). Calculating the value of this point is quite simple.

For example, for , the value makes the absolute value equal to 0.

Examples of others are:

The graph of an absolute value equation will be a V-shape that opens upward for any positive absolute function and opens downward for any negative absolute value function.

Example 4.4.2

Plot the graph of

The data point that gives the -value of the vertex is in which This is the first value.

For which yields

Now pick -values that are larger and less than −2 to get three data points on both sides of the vertex,

1	0
0	−1
−1	−2
−2	−3
−3	−2
−4	−1
−5	0

Once there are three data points on either side of the vertex, plot and connect them in a line. The graph is complete.

Example 4.4.3

Plot the graph of

The data point that gives the -value of the vertex is in which This is the first value.

For which yields

Now pick -values that are larger and less than 2 to get three data points on both sides of the vertex,

5	−2
4	−1
3	0
2	1
1	0
0	−1
−1	−2

Once there are three data points on either side of the vertex, plot and connect them in a line. The graph is complete.

Questions

For questions 1 to 8, graph each linear inequality.

1. 3x + 2" title="Rendered by QuickLaTeX.com" height="16" width="82" style="vertical-align: -4px;">
2. 12" title="Rendered by QuickLaTeX.com" height="16" width="99" style="vertical-align: -4px;">
3. 
4. 
5. 5x + 10" title="Rendered by QuickLaTeX.com" height="17" width="100" style="vertical-align: -4px;">
6. -20" title="Rendered by QuickLaTeX.com" height="17" width="114" style="vertical-align: -4px;">
7. 
8. 

For questions 9 to 16, graph each absolute value equation.

Answer Key 4.4

It is common to run into geometry-based word problems that look at either the interior angles, perimeter, or area of shapes. When looking at interior angles, the sum of the angles of any polygon can be found by taking the number of sides, subtracting 2, and then multiplying the result by 180°. In other words:

This means the interior angles of a triangle add up to 180° × (3 − 2), or 180°. Any four-sided polygon will have interior angles adding to 180° × (4 − 2), or 360°. A chart can be made of these:

Common Geometric Shapes with Related Area and Perimeter Equations

Shape	Picture	Area	Perimeter
Circle
Square
Rectangle
Triangle
Rhombus
Trapezoid
Parallelogram
Regular polygon (-gon)

Another common geometry word problem involves perimeter, or the distance around an object. For example, consider a rectangle, for which

Other common geometric problems are:

Questions

For questions 1 to 8, write the formula defining each relation. Do not solve.

1. The length of a rectangle is 3 cm less than double the width, and the perimeter is 54 cm.
2. The length of a rectangle is 8 cm less than double its width, and the perimeter is 64 cm.
3. The length of a rectangle is 4 cm more than double its width, and the perimeter is 32 cm.
4. The first angle of a triangle is twice as large as the second and 10° larger than the third.
5. The first angle of a triangle is half as large as the second and 20° larger than the third.
6. The sum of the first and second angles of a triangle is half the amount of the third angle.
7. A 140 cm cable is cut into two pieces. The first piece is five times as long as the second.
8. A 48 m piece of hose is to be cut into two pieces such that the second piece is 5 m longer than the first.

For questions 9 to 18, write and solve the equation describing each relationship.

9. The second angle of a triangle is the same size as the first angle. The third angle is 12° larger than the first angle. How large are the angles?
10. Two angles of a triangle are the same size. The third angle is 12° smaller than the first angle. Find the measure of the angles.
11. Two angles of a triangle are the same size. The third angle is three times as large as the first. How large are the angles?
12. The second angle of a triangle is twice as large as the first. The measure of the third angle is 20° greater than the first. How large are the angles?
13. Find the dimensions of a rectangle if the perimeter is 150 cm and the length is 15 cm greater than the width.
14. If the perimeter of a rectangle is 304 cm and the length is 40 cm longer than the width, find the length and width.
15. Find the length and width of a rectangular garden if the perimeter is 152 m and the width is 22 m less than the length.
16. If the perimeter of a rectangle is 280 m and the width is 26 m less than the length, find its length and width.
17. A lab technician cuts a 12 cm piece of tubing into two pieces such that one piece is two times longer than the other. How long are the pieces?
18. An electrician cuts a 30 m piece of cable into two pieces. One piece is 2 m longer than the other. How long are the pieces?

Answer Key 4.5

Cally, Katy, and Jasnah buy a bulk container of 24 litres of cooking oil. The problem is that they intend to divide it evenly into 8-litre amounts in three different containers. The other two containers are 13 litres and 11 litres. They have a small 5 litre container that can be used. Katy comes up with a solution, and they all agree that dividing the oil will require seven careful steps. Can you find Katy’s solution or find one that takes fewer steps?

Answer Key 4.6

The Eiffel Tower was completed in 1889 by Gustave Eiffel as the entrance for the world’s fair held in Paris the same year. It has since been visited by an estimated quarter billion people and remains the tallest structure in Paris with a height of 324 m.

Some of the particulars to this construction are that it is comprised of 18,038 pieces of metal held together by some 2.5 million rivets. There are 1,710 steps to the top of the tower, and the iron from which it was fashioned has a mass of 7300 tonnes and requires painting (60 tonnes) every seven years to prevent rusting. The top of the tower (the sun-facing side) can shift up to 18 cm due to thermal expansion. The density of the wrought iron used to create this tower is 7.70 g/cm³ and was allegedly sourced from Reșița in Romania.

Question: If one were to dismantle the Eiffel Tower and melt it to create an iron sphere, what would be its radius?

1. Evaluate: if and
2. Solve for in the equation
3. Isolate the variable in the equation .
4. Solve for in the equation .
5. Write an equation of the vertical line that passes through the point
6. Find the equation that has a slope of and passes through the point
7. Find the equation of the line passing through the points and .
8. Graph the relation .

For questions 9 to 11, find each solution set and graph it.

9. 67" title="Rendered by QuickLaTeX.com" height="18" width="154" style="vertical-align: -4px;">
10. 
11. 
12. Graph the relation
13. Find two numbers such that 5 times the larger number plus 3 times the smaller is 47, and 4 times the larger minus twice the smaller is 20.
14. Karl is going to cut a 36 cm cable into 2 pieces. If the first piece is to be 5 times as long as the second piece, find the length of each piece.
15. varies jointly with and and inversely with the square of If when and find the constant then use to find when and

Midterm 1: Version A Answer Key

1. Evaluate: if and
2. Solve for in the equation
3. Isolate the variable in the equation
4. Solve for in the equation
5. Find the equation of the horizontal line that passes through the point
6. Find the equation that has a slope of and passes through the point
7. Find the equation of the line passing through the points and
8. Graph the relation

For questions 9 to 11, find each solution set and graph it.

9. 5 (3 - 4x) + 7" title="Rendered by QuickLaTeX.com" height="18" width="239" style="vertical-align: -4px;">
10. 
11. 
12. Graph the relation
13. Find two consecutive even integers such that their sum is 16 less than five times the first integer.
14. Karl is going to cut a 40 cm cable into 2 pieces. If the first piece is to be 4 times as long as the second piece, find the length of each piece.
15. varies directly as and inversely as If when and find the constant then use this to find when and

Midterm 1: Version B Answer Key

1. Evaluate if and
2. Solve for in the equation
3. Isolate the variable in the equation
4. Solve for in the equation
5. Write the equation of the vertical line that passes through the point
6. Find the equation that has a slope of and passes through the point
7. Find the equation of the line passing through the points and
8. Graph the relation

For questions 9 to 11, find each solution set and graph it.

9. 
10. 3\text{ or }y-1<-3" title="Rendered by QuickLaTeX.com" height="16" width="186" style="vertical-align: -4px;">
11. 5" title="Rendered by QuickLaTeX.com" height="18" width="89" style="vertical-align: -4px;">
12. Graph the relation
13. Find two numbers such that 5 times the larger number plus 3 times the smaller is 49, and 4 times the larger minus twice the smaller is 26.
14. Karl is going to cut a 42 cm cable into 2 pieces. If the first piece is to be 5 times as long as the second piece, find the length of each piece.
15. varies jointly with and inversely with the square of If when and find the constant then use to find when and

Midterm 1: Version C Answer Key

1. Evaluate: if and
2. Solve for in the equation
3. Isolate the variable in the equation
4. Solve for in the equation
5. Write the equation of the horizontal line that passes through the point
6. Find the equation that has a slope of and passes through the point
7. Find the equation of the line passing through the points and
8. Graph the relation

For questions 9 to 11, find each solution set and graph it.

9. 
10. 
11. 6" title="Rendered by QuickLaTeX.com" height="18" width="90" style="vertical-align: -4px;">
12. Graph the relation
13. For a given triangle, the first and second angles are equal, but the third angle is 10° less than twice the first angle. What are the measures of the three angles?
14. Find two consecutive even integers such that their sum is 20 less than the first integer.
15. varies jointly with and the square of and inversely with If when and find the constant then use to find when and

Midterm 1: Version D Answer Key

1. Simplify the following:
   1. 
   2. 
   3. 
   4. 
   5. 

2. Solve for in the equation
3. Isolate the variable in the equation
4. Solve for in the equation
5. Find the equation of the horizontal line that passes through the point
6. Find the equation that has a slope of and passes through the point
7. Find the equation of the line passing through the points and
8. Graph the relation

For questions 9 to 11, find each solution set and graph it.

9. 
10. 
11. 1" title="Rendered by QuickLaTeX.com" height="44" width="95" style="vertical-align: -17px;">
12. Graph
13. Find three consecutive odd integers such that the sum of the first integer, two times the second integer, and three times the third integer is 94.
14. Karl is going to cut a 800 cm cable into 2 pieces. If the first piece is to be 3 times as long as the second piece, find the length of each piece.
15. varies jointly with and inversely with the square of If when and find the constant then use to find when and

Midterm 1: Version E Answer Key

Often, it is necessary to find the coordinates that are shared by two or more equations. There are multiple methods to find these shared values. While not giving values that are precise, graphing these equations make it possible to see the approximate solution and what type of solution it is.

Problems like have been solved in this textbook by adding 4 to both sides and then dividing by 3 (solution is ). There are also methods to solve equations with more than one variable in them. It turns out that, to solve for more than one variable, it is necessary to have the same number of equations as variables. For example, to solve for two variables such as and two equations are required. When there are several equations that must be solved, that is called a system of equations. When solving a system of equations, the solution must work in both equations. This solution is usually given as an ordered pair, The following example illustrates a solution working in both equations.

Note that the graph of an equation yields a picture of all its solutions. When two equations are graphed on the same coordinate plane, this displays not only the solutions of both equations, but where these solutions intersect. For instance, these solutions will show if the linear equations intersect at a point, over a line, or not at all.

In the following examples, each of the three possible types of solutions will be explored.

The coordinate is the shared point between both linear equations. This type of intersection is a unique solution that is called consistent and independent.

It is also possible to have a situation in which the same linear equation is graphed twice. Such an equation is easy to create.

Take the equation :

These are all the same equation, and if any two of them were graphed, the result would be the exact same line. This type of intersection has many solutions and is called consistent and dependent.

The two lines from the previous example have the exact same intercepts and, when graphed, draw the exact same line. Since the two graphs have solutions, it is defined as being consistent. However, the many solutions means they are dependent.

Lines that are parallel will never intersect and, as a result, will never have a solution or a shared intersection. Any system of equations having no solution is defined as being inconsistent. Parallel equations are identical except for having different intercepts. This means that the equation is parallel to:

Each of the above equations is parallel and will never intersect with each other.

Questions

For questions 1 to 12, find the intersection point of each system of equations.

Chapter 5.1 Answer Key

While solving a system by graphing has advantages, it also has several limitations. First, it requires the graph to be perfectly drawn: if the lines are not straight, it may result in the wrong answer. Second, graphing is challenging if the values are really large—over 100, for example—or if the answer is a decimal that the graph will not be able to depict accurately, like 3.2134. For these reasons, graphing is rarely used to solve systems of equations. Commonly, algebraic approaches such as substitution are used instead.

Questions

For questions 1 to 20, solve each system of equations by substitution.

Answer Key 5.2

One of the most powerful methods for solving systems of equations (finding their intersection points) is in adding and subtracting equations. In later math courses, this process is the foundation of matrix algebra, but for now, consider only equations.

The objective in finding the solutions to the these systems of equations is to isolate variables and find what they are equal to. Adding and subtracting equations can make this process quite fast and easy.

Generally, it takes a little more work than just placing the equations on top of each other and having a variable cancel out. In most cases, there is typically one variable that, once multiplied, is cancelled out when the equations are added to each other. For instance:

The more difficult of systems of two linear equations generally require the manipulation of both equations to eliminate one of the variables. For example, consider the following pair of linear equations:

The last examples that will be done for this topic are equations having no solution or infinite solutions.

Questions

For questions 1 to 24, solve each system of equations by elimination.

Answer Key 5.3

When given three variables, you are given the equation for a plane or a flat surface similar to a sheet of paper. Some of the possible solutions to the intersections of these equations can be visualized below.

In solving systems of equations with three variables, use the strategies that are used to solve systems of two equations. One recommended method is to eliminate one variable at the onset, thus turning the set of three equations with three unknowns into two equations with two unknowns. The standard method to work with three equations or more is to use subtraction and/or addition.

Sometimes, you are given a set of three equations with missing variables. These systems of equations require slightly more thought to solve than the previous problems.

Questions

Solve each of the following systems of equations.

Answer Key 5.4

Solving value problems generally involves the solution of systems of equations. Value problems are ones in which each variable has a value attached to it, such as a nickel being worth 5¢, a dollar worth and a stamp worth 85¢. Using a table will help to set up and solve these problems. The basic structure of this table is shown below:

The first column in the table (Name) is used to identify the objects in the problem. The second column (Amount) identifies the amounts of each of the objects. The third column (Value) is used for the value of each object. The last column (Equation) is the product of the Amount times the Value.

Example 5.5.1

In Cally’s piggy bank, there are 11 coins having a value of Each coin is either a quarter or a dime. Use this data to fill in the chart and find the equation to be solved.

• The objects’ names are quarters (Q) and dimes (D)
• which means or
• Quarters have a value of and dimes have a value of

Name	Amount	Value	Equation
Quarters (Q)
Dimes (D)
Mixture	11	N/A

The first two equations generally combine to equal the third equation.

The equation derived from this data is:

Example 5.5.2

Doug and Becky sold 41 tickets for an event. Tickets for children cost and tickets for adults cost . Total receipts for the event were How many of each type of ticket was sold?

Name	Amount	Value	Equation
Children (C)
Adult (A)
Mixture	41	N/A

The equation to be solved is:

This means that the number of adult tickets sold was

Example 5.5.3

Nick has a collection of 5-cent stamps and 8-cent stamps. There are three times as many 8-cent stamps as 5-cent stamps. The total value of all the stamps is How many of each stamp does Nick have?

Name	Amount	Value	Equation
Five-centers (F)
Eight-centers (E)
Mixture		N/A

The equation to be solved is:

This means that the number of eight-cent stamps is

Example 5.5.4

Angela invests in two accounts, one at 6% interest, the other at 9% interest for one year. At the end of the year, she had earned in interest. How much did she have invested in each account?

Name	Amount	Value	Equation
Account (S)		0.06
Account (N)		0.09
Total interest		N/A

The equation to be solved is:

This means that the amount in the nine-percent account is

Example 5.5.5

Clark and Kyra invest in one account and in an account paying 4% more in interest. They earned in interest after one year. At what rates did they invest?

Name	Amount	Value	Equation
Account (F)
Account (E)
Total interest	N/A	N/A

The equation to be solved is:

The other interest rate is

This means that was invested at 7% and was invested at 11%.

Questions

For questions 1 to 10, find the equations that describe each problem. Do not solve.

1. A collection of dimes and quarters is worth There are 103 coins in all. How many of each kind of coin is there?
2. A collection of fifty-cent pieces and nickels is worth There are 34 coins in all. How many of each kind of coin is there?
3. The attendance at a school concert was 578. Admission was for adults and for children. The total of the receipts was How many adults and how many children attended?
4. Natasha’s purse contains made up of dimes and quarters. If there are 21 coins in all, how many dimes and how many quarters are there?
5. A boy has in nickels and dimes. If there are twice as many dimes as nickels, how many of each kind of coin does he have?
6. is made up of quarters and fifty-cent pieces. If the number of quarters exceeds the number of fifty-cent pieces by three, how many coins of each denomination are there?
7. An inheritance of is invested in two ways, part at 9.5% and the remainder at 11%. The combined annual interest was How much was invested at each rate?
8. Kerry earned a total of last year on his investments. If was invested at a certain rate of return and was invested in a fund with a rate that was 2% higher, find the two rates of interest.
9. Jason earned in interest last year on his investments. If was invested at a certain rate of return and was invested in a fund with a rate that was double the rate of the first fund, find the two rates of interest.
10. Millicent earned last year in interest. If was invested at a certain rate of return and was invested in a fund with a rate that was 2% lower, find the two rates of interest.

For questions 11 to 25, find and solve the equations that describe each problem.

11. There were 203 tickets sold for a volleyball game. For activity-card holders, the price was each, and for non-card holders, the price was each. The total amount of money collected was How many of each type of ticket was sold?
12. At a local ball game, the hot dogs sold for each and the hamburgers sold for each. There were 131 total food items sold for a total value of How many of each item was sold?
13. A piggy bank contains 27 total dimes and quarters. The coins have a total value of Find the number of dimes and quarters in the piggy bank.
14. A coin purse contains 18 total nickels and dimes. The coins have a total value of Find the number of nickels and dimes in the coin purse.
15. Sally bought 40 stamps for The purchase included 25¢ stamps and 20¢ stamps. How many of each type of stamp was bought?
16. A postal clerk sold some 15¢ stamps and some 25¢ stamps. Altogether, 15 stamps were sold for a total cost of How many of each type of stamp was sold?
17. A total of is invested, part of it at 12% and the rest at 13%. The total interest after one year is How much was invested at each rate?
18. A total of is invested, part of it at 5% and the rest at 7.5%. The total interest after one year is How much was invested at each rate?
19. The total value of dimes and quarters in a piggy bank is There are six more quarters than dimes. Find the number of each type of coin in the piggy bank.
20. A piggy bank contains nickels and dimes. The number of dimes is ten less than twice the number of nickels. The total value of all the coins is Find the number of each type of coin in the piggy bank.
21. An investment portfolio earned in interest last year. If was invested at a certain rate of return and was invested in a fund with a rate that was 4% lower, find the two rates of interest.
22. Samantha earned in interest last year on their investments. If was invested at a certain rate of return and was invested in a fund with a rate that was two-thirds the rate of the first fund, find the two rates of interest.
23. A man has in nickels, dimes, and quarters. There are twice as many nickels as dimes and three more dimes than quarters. How many coins of each kind are there?
24. A bag containing nickels, dimes, and quarters has a value of If there are 40 coins in all and three times as many dimes as quarters, how many coins of each kind are there?
25. A collection of stamps consists of 22¢ stamps and 40¢ stamps. The number of 22¢ stamps is three more than four times the number of 40¢ stamps. The total value of the stamps is Find the number of 22¢ stamps in the collection.

Answer Key 5.5

Three-dimensional space and time problems. Have fun solving these!  🙂

Questions

Answer Key 5.6

The internet has many of the following picture number puzzles. All are quickly and easily solved with the strategies used for solving systems of equations. What are the solutions for the following picture puzzles?

1. 
2. 
3. A more challenging system of equations is the four-by-four puzzle shown below. Each individual row and column are summed to the number in either row or column 5. Find the five missing numbers and place them in their correct positions.

Rows	Column 1	Column 2	Column 3	Column 4	Column 5
Row 1					?
Row 2
Row 3					?
Row 4
Row 5		?		?	?

Answer key 5.7

Exponents often can be simplified using a few basic properties, since exponents represent repeated multiplication. The basic structure of writing an exponent looks like where is defined as the base and is termed its exponent. For this instance, represents the number of times that the variable is multiplied by itself

When looking at numbers to various powers, the following table gives the numeric value of several numbers to various powers.

For this chart, the expanded forms of the base 2 for multiple exponents is shown:

Once there is an exponent as a base that is multiplied or divided by itself to the number represented by the exponent, it becomes straightforward to identify a number of rules and properties that can be defined.

The following examples outline a number of these rules.

Questions

Simplify the following.

Answer Key 6.1

Consider the following chart that shows the expansion of for several exponents:

If zero and negative exponents are expanded to base 2, the result is the following:

The most unusual of these is the exponent 0. Any base that is not equal to zero to the zeroth exponent is always 1. The simplest explanation of this is by example.

This zero rule of exponents can make difficult problems elementary simply because whatever the 0 exponent is attached to reduces to 1. Consider the following examples:

When encountering these types of problems, always remain aware of what the zero power is attached to, since only what it is attached to cancels to 1.

When dealing with negative exponents, the simplest solution is to reciprocate the power. For instance:

Four Rules of Negative Exponents

Questions

Simplify. Your answer should contain only positive exponents.

Answer Key 6.2

Scientific notation is a convenient notation system used to represent large and small numbers. Examples of these are the mass of the sun or the mass of an electron in kilograms. Simplifying basic operations such as multiplication and division with these numbers requires using exponential properties.

Scientific notation has two parts: a number between one and nine and a power of ten, by which that number is multiplied.

The exponent tells how many times to multiply by 10. Each multiple of 10 shifts the decimal point one place. To decide which direction to move the decimal (left or right), recall that positive exponents means there is big number (larger than ten) and negative exponents means there is a small number (less than one).

Working with scientific notation is easier than working with other exponential notation, since the base of the exponent is always 10. This means that the exponents can be treated separately from any other numbers. For instance:

Questions

For questions 1 to 6, write each number in scientific notation.

1. 885.3
2. 0.000744
3. 0.081
4. 1.09
5. 0.039
6. 15,000

For questions 7 to 12, write each number in standard notation.

7. 8.7 × 10⁵
8. 2.56 × 10²
9. 9 × 10⁻⁴
10. 5 × 10⁴
11. 2 × 10⁰
12. 6 × 10⁻⁵

For questions 13 to 20, simplify each expression and write each answer in scientific notation.

13. (7 × 10⁻¹)(2 × 10⁻³)
14. (2 × 10⁻⁶)(8.8 × 10⁻⁵)
15. (5.26 × 10⁻⁵)(3.16 × 10⁻²)
16. (5.1 × 10⁶)(9.84 × 10⁻¹)
17. 
18. 
19. 
20. 

Answer Key 6.3

Many applications in mathematics have to do with what are called polynomials. Polynomials are made up of terms. Terms are a product of numbers and/or variables. For example, , , , , and are all terms. Terms are connected to each other by addition or subtraction.

Expressions are often defined by the number of terms they have.

The term polynomial is generic for many terms. Monomials, binomials, trinomials, and expressions with more terms all fall under the umbrella of “polynomials.”

Polynomials are classified by the sum of exponents of the term with the highest exponent sum, which is called the degree of the polynomial.

A first degree polynomial is a linear polynomial and would not have any terms with a sum of exponents greater than one. Examples include , , .

A second degree polynomial is a quadratic polynomial and would not have any terms with a sum of exponents greater than two. Examples include , , .

A third degree polynomial is a cubic polynomial and would not have any terms with a sum of exponents greater than three. Examples include , , .

A fourth degree polynomial is a quartic polynomial and would not have any terms with a sum of exponents greater than four. Examples include , , .

A fifth degree polynomial is a quintic polynomial.

A sixth degree polynomial is a sextic polynomial.

A seventh degree polynomial is a septic polynomial.

A eighth degree polynomial is a octic polynomial.

A ninth degree polynomial is a nonic polynomial.

A tenth degree polynomial is a decic polynomial.

The degree of any term is the sum of its exponents:

are all ninth degree terms (nonic polynomial)

are all sixth degree terms (sextic polynomial)

are all fourth degree terms (quartic polynomial)

Terms of a polynomial are named in the order of their appearance. For instance, the polynomial has four terms, each one of the fifth degree. The terms are numbered in order for this polynomial, starting from the first term and continuing to the second ((x^3y^2)\), third , and fourth terms .

If it is known what the variable in a polynomial represents, it is possible to substitute in the value and evaluate the polynomial, as shown in the following example.

Remember the exponent only affects the number to which it is physically attached. This means −3² = −9 because the exponent is only attached to the 3, whereas (−3)² = 9 because the exponent is attached to the parentheses and everything inside.

Sometimes, when working with polynomials,  the value of the variable is unknown and the polynomial will be simplified rather than ending with some value. The simplest operation with polynomials is addition. When adding polynomials, it is merely combining like terms. Consider the following example.

Subtracting polynomials is almost as fast as addition. Subtraction is generally shown by a minus sign in front of the parentheses. When there is a negative sign in front of parentheses,  distribute it throughout the expression, changing the signs of everything inside.

Questions

For questions 1 to 8, simplify each expression using the variables given.

For questions 9 to 28, simplify the following expressions.

Answer Key 6.4

Multiplying monomials is done by multiplying the numbers or coefficients and then adding the exponents on like factors. This is shown in the next example.

Some notes: has an exponent of 1 when no exponent is written. When adding or subtracting, the exponents will stay the same, but when multiplying (or dividing), the exponents will change.

Next,  consider multiplying a monomial by a polynomial. We have seen this operation before with distributing throughout parentheses. Here, its the exact same process.