# Integers

## Solve Equations Using Integers; The Division Property of Equality


### Determine Whether a Number is a Solution of an Equation

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations. 
The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

### Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.
When you add or subtract the same quantity from both sides of an equation, you still have equality.

### Model the Division Property of Equality

All of the equations we have solved so far have been of the form  or  We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.
We will model an equation with envelopes and counters in .
Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
To determine the number, separate the counters on the right side into  groups of the same size. So  counters divided into  groups means there must be  counters in each group (since 
What equation models the situation shown in ? There are two envelopes, and each contains  counters. Together, the two envelopes must contain a total of  counters. So the equation that models the situation is 
We can divide both sides of the equation by  as we did with the envelopes and counters.
We found that each envelope contains  Does this check? We know  so it works. Three counters in each of two envelopes does equal six.
 shows another example.
Now we have  identical envelopes and  How many counters are in each envelope? We have to separate the  into  Since  there must be  in each envelope. See .
The equation that models the situation is  We can divide both sides of the equation by 
Does this check? It does because 

### Solve Equations Using the Division Property of Equality

The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.

### Translate to an Equation and Solve

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

### Key Concepts

1. How to determine whether a number is a solution to an equation.


2. Properties of Equalities




3. Division Property of Equality




### Section Exercises


### Practice Makes Perfect

Determine Whether a Number is a Solution of an Equation 
In the following exercises, determine whether each number is a solution of the given equation.
Solve Equations Using the Addition and Subtraction Properties of Equality 
In the following exercises, solve for the unknown.
Model the Division Property of Equality 
In the following exercises, write the equation modeled by the envelopes and counters and then solve it.
Solve Equations Using the Division Property of Equality 
In the following exercises, solve each equation using the division property of equality and check the solution.
Translate to an Equation and Solve 
In the following exercises, translate and solve.
Mixed Practice 
In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


### Chapter Review Exercises


### Introduction to Integers

Locate Positive and Negative Numbers on the Number Line 
In the following exercises, locate and label the integer on the number line.
Order Positive and Negative Numbers 
In the following exercises, order each of the following pairs of numbers, using  or 
Find Opposites 
In the following exercises, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, evaluate.
Simplify Absolute Values 
In the following exercises, simplify.
In the following exercises, evaluate.
In the following exercises, fill in  for each of the following pairs of numbers.
In the following exercises, simplify.
Translate Phrases to Expressions with Integers 
In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.

### Add Integers

Model Addition of Integers 
In the following exercises, model the following to find the sum.
Simplify Expressions with Integers 
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers 
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions 
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Add Integers in Applications 
In the following exercises, solve.

### Subtract Integers

Model Subtraction of Integers 
In the following exercises, model the following.
Simplify Expressions with Integers 
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers 
In the following exercises, evaluate each expression.
Translate Phrases to Algebraic Expressions 
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Subtract Integers in Applications 
In the following exercises, solve the given applications.

### Multiply and Divide Integers

Multiply Integers 
In the following exercises, multiply.
Divide Integers 
In the following exercises, divide.
Simplify Expressions with Integers 
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers 
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions 
In the following exercises, translate to an algebraic expression and simplify if possible.

### Solve Equations using Integers; The Division Property of Equality

Determine Whether a Number is a Solution of an Equation 
In the following exercises, determine whether each number is a solution of the given equation.
Using the Addition and Subtraction Properties of Equality 
In the following exercises, solve.
Model the Division Property of Equality 
In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.
Solve Equations Using the Division Property of Equality 
In the following exercises, solve each equation using the division property of equality and check the solution.
Translate to an Equation and Solve. 
In the following exercises, translate and solve.

### Everyday Math


### Chapter Practice Test

In the following exercises, compare the numbers, using 
In the following exercises, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, evaluate.
In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.
In the following exercises, solve.
In the following exercises, solve.
In the following exercises, translate and solve.