# Solving Linear Equations and Inequalities

## Solve Linear Inequalities


### Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
What about the solution of an inequality? What number would make the inequality  true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality .
We show the solutions to the inequality  on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of  is shown in . Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.
The graph of the inequality  is very much like the graph of , but now we need to show that 3 is a solution, too. We do that by putting a bracket at , as shown in .
Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.
We can also represent inequalities using interval notation. As we saw above, the inequality  means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express  as  The symbol  is read as ‘infinity’. It is not an actual number.  shows both the number line and the interval notation.
The inequality  means all numbers less than or equal to 1. There is no lower end to those numbers. We write  in interval notation as . The symbol  is read as ‘negative infinity’.  shows both the number line and interval notation.
Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in .

### Solve Inequalities using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.
Similar properties hold true for inequalities.
Similarly we could show that the inequality also stays the same for addition.
This leads us to the Subtraction and Addition Properties of Inequality.
We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality , the steps would look like this:
Any number greater than 4 is a solution to this inequality.

### Solve Inequalities using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).
Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?
Consider some numerical examples.
Does the inequality stay the same when we divide or multiply by a negative number?
When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.
Here are the Division and Multiplication Properties of Inequality for easy reference.
When we divide or multiply an inequality by a:
1. positive number, the inequality stays the same.
2. negative number, the inequality reverses.

### Solve Inequalities That Require Simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.
Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

### Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.
Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.
 shows some common phrases that indicate inequalities.

### Key Concepts

1. Subtraction Property of Inequality
For any numbers a, b, and c,
if  then  and
if  then 
2. Addition Property of Inequality
For any numbers a, b, and c,
if  then  and
if  then 
3. Division and Multiplication Properties of Inequality
For any numbers a, b, and c,
if  and , then  and .
if  and , then  and .
if  and , then  and .
if  and , then  and .
4. When we divide or multiply an inequality by a:




### Section Exercises


### Practice Makes Perfect

Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Inequalities using the Subtraction and Addition Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities using the Division and Multiplication Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities That Require Simplification
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Mixed practice
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate to an Inequality and Solve
In the following exercises, translate and solve .Then write the solution in interval notation and graph on the number line.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter 2 Review Exercises


### Solve Equations using the Subtraction and Addition Properties of Equality

Verify a Solution of an Equation
In the following exercises, determine whether each number is a solution to the equation.
Solve Equations using the Subtraction and Addition Properties of Equality
In the following exercises, solve each equation using the Subtraction Property of Equality.
In the following exercises, solve each equation using the Addition Property of Equality.
In the following exercises, solve each equation.
Solve Equations That Require Simplification
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
Translate and Solve Applications
In the following exercises, translate into an algebraic equation and solve.

### Solve Equations using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation using the division and multiplication properties of equality and check the solution.
Solve Equations That Require Simplification
In the following exercises, solve each equation requiring simplification.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.

### Solve Equations with Variables and Constants on Both Sides

Solve an Equation with Constants on Both Sides
In the following exercises, solve the following equations with constants on both sides.
Solve an Equation with Variables on Both Sides
In the following exercises, solve the following equations with variables on both sides.
Solve an Equation with Variables and Constants on Both Sides
In the following exercises, solve the following equations with variables and constants on both sides.

### Use a General Strategy for Solving Linear Equations

Solve Equations Using the General Strategy for Solving Linear Equations
In the following exercises, solve each linear equation.
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

### Solve Equations with Fractions and Decimals

Solve Equations with Fraction Coefficients
In the following exercises, solve each equation with fraction coefficients.
Solve Equations with Decimal Coefficients
In the following exercises, solve each equation with decimal coefficients.

### Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula
In the following exercises, solve.
Solve a Formula for a Specific Variable
In the following exercises, solve.

### Solve Linear Inequalities

Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Inequalities using the Subtraction and Addition Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities using the Division and Multiplication Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities That Require Simplification
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate to an Inequality and Solve
In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.

### Everyday Math


### Chapter 2 Practice Test

In the following exercises, solve each equation.
In the following exercises, graph on the number line and write in interval notation.
In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
In the following exercises, translate to an equation or inequality and solve.