Solving Linear Equations and Inequalities
23 Solve Linear Inequalities
Learning Objectives
By the end of this section, you will be able to:
 Graph inequalities on the number line
 Solve inequalities using the Subtraction and Addition Properties of inequality
 Solve inequalities using the Division and Multiplication Properties of inequality
 Solve inequalities that require simplification
 Translate to an inequality and solve
Before you get started, take this readiness quiz.
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 (Figure). 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 (Figure).
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.
Graph on the number line:
ⓐⓑⓒ
 ⓐ
This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at to show that it is included.
 ⓑ
This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at to show it is not included.
 ⓒ
This means all numbers greater than , but not including . We shade in all the numbers on the number line to the right of , then put a parenthesis at to show it is not included.
Graph on the number line: ⓐ ⓑ ⓒ
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Graph on the number line: ⓐ ⓑ ⓒ
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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. (Figure) 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’. (Figure) 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 (Figure).
Graph on the number line and write in interval notation.
ⓐⓑⓒ
 ⓐ
Shade to the right of , and put a bracket at . Write in interval notation.
ⓑ
Shade to the left of , and put a parenthesis at . Write in interval notation.
ⓒ
Shade to the left of , and put a bracket at . Write in interval notation.
Graph on the number line and write in interval notation:
ⓐⓑⓒ
 ⓐ
 ⓑ
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Graph on the number line and write in interval notation:
ⓐⓑⓒ
 ⓐ
 ⓑ
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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.
For example, we know that −4 is less than 2.  
If we subtract 5 from both quantities, is the left side still less than the right side? 

We get −9 on the left and −3 on the right.  
And we know −9 is less than −3.  
The inequality sign stayed the same. 
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:
Subtract 5 from both sides to isolate .  
Simplify. 
Any number greater than 4 is a solution to this inequality.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Add to both sides of the inequality.  
Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Solve the 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
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.
Divide both sides by 5.  Multiply both sides by 5.  
Simplify.  
Fill in the inequality signs. 
Does the inequality stay the same when we divide or multiply by a negative number?
Divide both sides by −5.  Multiply both sides by −5.  
Simplify.  
Fill in the inequality signs. 
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:
 positive number, the inequality stays the same.
 negative number, the inequality reverses.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Divide both sides of the inequality by 7. Since , the inequality stays the same. 

Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Divide both sides of the inequality by −10. Since , the inequality reverses. 

Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.
Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Multiply both sides of the inequality by . Since , the inequality stays the same. 

Simplify.  
Rewrite the variable on the left.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Multiply both sides of the inequality by . Since , the inequality reverses. 

Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
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.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Subtract from both sides to collect the variables on the left.  
Simplify.  
Divide both sides of the inequality by −5, and reverse the inequality.  
Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Simplify each side as much as possible.  
Distribute.  
Combine like terms.  
Subtract from both sides to collect the variables on the left.  
Simplify.  
Add 36 to both sides to collect the constants on the right.  
Simplify.  
Divide both sides of the inequality by 4; the inequality stays the same.  
Simplify.  
Graph the solution on the number line.  
Write the solution in interal notation. 
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
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.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Simplify each side as much as possible.  
Distribute.  
Combine like terms.  
Subtract from both sides to collect the variables on the left.  
Simplify.  
The ’s are gone, and we have a true statement.  The inequality is an identity. The solution is all real numbers. 
Graph the solution on the number line.  
Write the solution in interval notation. 
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Multiply both sides by the LCD, 24, to clear the fractions.  
Simplify.  
Combine like terms.  
Subtract from both sides to collect the variables on the left.  
Simplify.  
The statement is false!  The inequality is a contradiction. 
There is no solution.  
Graph the solution on the number line.  
Write the solution in interval notation.  There is no solution. 
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
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’.
(Figure) shows some common phrases that indicate inequalities.
is greater than  is greater than or equal to  is less than  is less than or equal to 
is more than  is at least  is smaller than  is at most 
is larger than  is no less than  has fewer than  is no more than 
exceeds  is the minimum  is lower than  is the maximum 
Translate and solve. Then write the solution in interval notation and graph on the number line.
Twelve times c is no more than 96.
Translate.  
Solve—divide both sides by 12.  
Simplify.  
Write in interval notation.  
Graph on the number line. 
Translate and solve. Then write the solution in interval notation and graph on the number line.
Twenty times y is at most 100
Translate and solve. Then write the solution in interval notation and graph on the number line.
Nine times z is no less than 135
Translate and solve. Then write the solution in interval notation and graph on the number line.
Thirty less than x is at least 45.
Translate.  
Solve—add 30 to both sides.  
Simplify.  
Write in interval notation.  
Graph on the number line. 
Translate and solve. Then write the solution in interval notation and graph on the number line.
Nineteen less than p is no less than 47
Translate and solve. Then write the solution in interval notation and graph on the number line.
Four more than a is at most 15.
Key Concepts
 Subtraction Property of Inequality
For any numbers a, b, and c,
if then and
if then  Addition Property of Inequality
For any numbers a, b, and c,
if then and
if then  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 .  When we divide or multiply an inequality by a:
 positive number, the inequality stays the same.
 negative number, the inequality reverses.
Section Exercises
Practice Makes Perfect
Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
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ⓐ
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ⓐ
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ⓐ
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In the following exercises, graph each inequality on the number line and write in interval notation.
ⓐ
ⓑ
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ⓐ
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ⓐ
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ⓐ
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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.
Fourteen times d is greater than 56.
Ninety times c is less than 450.
Eight times z is smaller than .
Ten times y is at most .
Three more than h is no less than 25.
Six more than k exceeds 25.
Ten less than w is at least 39.
Twelve less than x is no less than 21.
Negative five times r is no more than 95.
Negative two times s is lower than 56.
Nineteen less than b is at most .
Fifteen less than a is at least .
Everyday Math
Safety A child’s height, h, must be at least 57 inches for the child to safely ride in the front seat of a car. Write this as an inequality.
Fighter pilots The maximum height, h, of a fighter pilot is 77 inches. Write this as an inequality.
Elevators The total weight, w, of an elevator’s passengers can be no more than 1,200 pounds. Write this as an inequality.
Shopping The number of items, n, a shopper can have in the express checkout lane is at most 8. Write this as an inequality.
Writing Exercises
Give an example from your life using the phrase ‘at least’.
Give an example from your life using the phrase ‘at most’.
Answers will vary.
Explain why it is necessary to reverse the inequality when solving .
Explain why it is necessary to reverse the inequality when solving .
Answers will vary.
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.
no
yes
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.
The sum of and is 25.
Four less than is 13.
Translate and Solve Applications
In the following exercises, translate into an algebraic equation and solve.
Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?
Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?
161 pounds
Peter paid ?9.75 to go to the movies, which was ?46.25 less than he paid to go to a concert. How much did he pay for the concert?
Elissa earned ?152.84 this week, which was ?21.65 more than she earned last week. How much did she earn last week?
?131.19
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.
143 is the product of and y.
The quotient of b and and 9 is .
The sum of q and onefourth is one.
The difference of s and onetwelfth is one fourth.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.
Ray paid ?21 for 12 tickets at the county fair. What was the price of each ticket?
Janet gets paid ?24 per hour. She heard that this is of what Adam is paid. How much is Adam paid per hour?
?32
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.
contradiction; no solution
identity; all real numbers
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.
Natalie drove for hours at 60 miles per hour. How much distance did she travel?
Mallory is taking the bus from St. Louis to Chicago. The distance is 300 miles and the bus travels at a steady rate of 60 miles per hour. How long will the bus ride be?
5 hours
Aaron’s friend drove him from Buffalo to Cleveland. The distance is 187 miles and the trip took 2.75 hours. How fast was Aaron’s friend driving?
Link rode his bike at a steady rate of 15 miles per hour for hours. How much distance did he travel?
37.5 miles
Solve a Formula for a Specific Variable
In the following exercises, solve.
Use the formula. to solve for t
ⓐ when and
ⓑ in general
Use the formula. to solve for
ⓐ when when and
ⓑ in general
ⓐ; ⓑ
Use the formula to solve for
ⓐ when and
ⓑ in general
Use the formula to solve for
ⓐ when and
ⓑ in general
ⓐⓑ
Use the formula to solve for the principal, P for
ⓐ
ⓑ in general
Solve the formula for y
ⓐ when
ⓑ in general
ⓐⓑ
Solve for .
Solve the formula for .
Solve Linear Inequalities
Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
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ⓑ
ⓒ
ⓐ
ⓑ
ⓒ
 ⓐ
 ⓑ
 ⓒ
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.
Five more than z is at most 19.
Three less than c is at least 360.
Nine times n exceeds 42.
Negative two times a is no more than 8.
Everyday Math
Describe how you have used two topics from this chapter in your life outside of your math class during the past month.
Chapter 2 Practice Test
Determine whether each number is a solution to the equation .
ⓐ 5
ⓑ
ⓐ no ⓑ yes
In the following exercises, solve each equation.
contradiction; no solution
Solve the formula for y
ⓐ when
ⓑ in general
ⓐⓑ
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.
4 less than twice x is 16.
Fifteen more than n is at least 48.
Samuel paid ?25.82 for gas this week, which was ?3.47 less than he paid last week. How much had he paid last week?
Jenna bought a coat on sale for ?120, which was of the original price. What was the original price of the coat?
; The original price was ?180.
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took hours, what was the speed of the bus?