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Representing and Solving Linear Inequalities: Worksheets with Answers

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2.7 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

Be Prepared 2.18

Before you get started, take this readiness quiz.

Translate from algebra to English: 15 > x 15 > x . If you missed this problem, review Example 1.12 .

Be Prepared 2.19

Solve: n − 9 = −42 . n − 9 = −42 . If you missed this problem, review Example 2.3 .

Be Prepared 2.20

Solve: −5 p = −23 . −5 p = −23 . If you missed this problem, review Example 2.13 .

Be Prepared 2.21

Solve: 3 a − 12 = 7 a − 20 . 3 a − 12 = 7 a − 20 . If you missed this problem, review Example 2.34 .

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 x > 3 x > 3 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 x > 3 x > 3 .

We show the solutions to the inequality x > 3 x > 3 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 x > 3 x > 3 is shown in Figure 2.7 . Please note that the following convention is used: thick arrows point in the positive direction and thin arrows point in the negative direction.

The graph of the inequality x ≥ 3 x ≥ 3 is very much like the graph of x > 3 x > 3 , but now we need to show that 3 is a solution, too. We do that by putting a bracket at x = 3 x = 3 , as shown in Figure 2.8 .

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.

Example 2.66

Graph on the number line:

ⓐ x ≤ 1 x ≤ 1 ⓑ x < 5 x < 5 ⓒ x > − 1 x > − 1

Try It 2.131

Graph on the number line: ⓐ x ≤ − 1 x ≤ − 1 ⓑ x > 2 x > 2 ⓒ x < 3 x < 3

Try It 2.132

Graph on the number line: ⓐ x > − 2 x > − 2 ⓑ x < − 3 x < − 3 ⓒ x ≥ −1 x ≥ −1

We can also represent inequalities using interval notation. As we saw above, the inequality x > 3 x > 3 means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation , we express x > 3 x > 3 as ( 3 , ∞ ) . ( 3 , ∞ ) . The symbol ∞ ∞ is read as ‘infinity’. It is not an actual number. Figure 2.9 shows both the number line and the interval notation.

The inequality x ≤ 1 x ≤ 1 means all numbers less than or equal to 1. There is no lower end to those numbers. We write x ≤ 1 x ≤ 1 in interval notation as ( − ∞ , 1 ] ( − ∞ , 1 ] . The symbol − ∞ − ∞ is read as ‘negative infinity’. Figure 2.10 shows both the number line and interval notation.

Inequalities, Number Lines, 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 2.11 .

Example 2.67

Graph on the number line and write in interval notation.

ⓐ x ≥ −3 x ≥ −3 ⓑ x < 2.5 x < 2.5 ⓒ x ≤ − 3 5 x ≤ − 3 5

Try It 2.133

Graph on the number line and write in interval notation:

ⓐ x > 2 x > 2 ⓑ x ≤ − 1.5 x ≤ − 1.5 ⓒ x ≥ 3 4 x ≥ 3 4

Try It 2.134

ⓐ x ≤ − 4 x ≤ − 4 ⓑ x ≥ 0.5 x ≥ 0.5 ⓒ x < − 2 3 x < − 2 3

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.

Properties of Equality

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.

Properties of Inequality

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality x + 5 > 9 x + 5 > 9 , the steps would look like this:

Any number greater than 4 is a solution to this inequality.

Example 2.68

Solve the inequality n − 1 2 ≤ 5 8 n − 1 2 ≤ 5 8 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.135

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

p − 3 4 ≥ 1 6 p − 3 4 ≥ 1 6

Try It 2.136

r − 1 3 ≤ 7 12 r − 1 3 ≤ 7 12

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.

Division and Multiplication Properties of Inequality

When we divide or multiply an inequality by a:

• positive number, the inequality stays the same .
• negative number, the inequality reverses .

Example 2.69

Solve the inequality 7 y < ​ ​ 42 7 y < ​ ​ 42 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.137

c - 8 > 0 c - 8 > 0

Try It 2.138

12 d ≤ ​ 60 12 d ≤ ​ 60

Example 2.70

Solve the inequality −10 a ≥ 50 −10 a ≥ 50 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.139

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

−8 q < 32 −8 q < 32

Try It 2.140

−7 r ≤ ​ − 70 −7 r ≤ ​ − 70

Solving Inequalities

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.”

Example 2.71

Solve the inequality −20 < 4 5 u −20 < 4 5 u , graph the solution on the number line, and write the solution in interval notation.

Try It 2.141

24 ≤ 3 8 m 24 ≤ 3 8 m

Try It 2.142

−24 < 4 3 n −24 < 4 3 n

Example 2.72

Solve the inequality t −2 ≥ 8 t −2 ≥ 8 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.143

k −12 ≤ 15 k −12 ≤ 15

Try It 2.144

u −4 ≥ −16 u −4 ≥ −16

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.

Example 2.73

Solve the inequality 4 m ≤ 9 m + 17 4 m ≤ 9 m + 17 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.145

Solve the inequality 3 q   ≥   7 q   −   23 3 q   ≥   7 q   −   23 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.146

Solve the inequality 6 x < 10 x + 19 6 x < 10 x + 19 , graph the solution on the number line, and write the solution in interval notation.

Example 2.74

Solve the inequality 8 p + 3 ( p − 12 ) > 7 p − 28 8 p + 3 ( p − 12 ) > 7 p − 28 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.147

Solve the inequality 9 y + 2 ( y + 6 ) > 5 y − 24 9 y + 2 ( y + 6 ) > 5 y − 24 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.148

Solve the inequality 6 u + 8 ( u − 1 ) > 10 u + 32 6 u + 8 ( u − 1 ) > 10 u + 32 , 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.

Example 2.75

Solve the inequality 8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x 8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x , graph the solution on the number line, and write the solution in interval notation.

Try It 2.149

Solve the inequality 4 b − 3 ( 3 − b ) > 5 ( b − 6 ) + 2 b 4 b − 3 ( 3 − b ) > 5 ( b − 6 ) + 2 b , graph the solution on the number line, and write the solution in interval notation.

Try It 2.150

Solve the inequality 9 h − 7 ( 2 − h ) < 8 ( h + 11 ) + 8 h 9 h − 7 ( 2 − h ) < 8 ( h + 11 ) + 8 h , graph the solution on the number line, and write the solution in interval notation.

Example 2.76

Solve the inequality 1 3 a − 1 8 a > 5 24 a ​ + 3 4 1 3 a − 1 8 a > 5 24 a ​ + 3 4 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.151

Solve the inequality 1 4 x − 1 12 x > 1 6 x + 7 8 1 4 x − 1 12 x > 1 6 x + 7 8 , graph the solution on the number line, and write the solution in interval notation.

Try It 2.152

Solve the inequality 2 5 z − 1 3 z < 1 15 z ​ − 3 5 2 5 z − 1 3 z < 1 15 z ​ − 3 5 , 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’.

Table 2.7 shows some common phrases that indicate inequalities.

Example 2.77

Translate and solve. Then write the solution in interval notation and graph on the number line.

Twelve times c is no more than 96.

Try It 2.153

Twenty times y is at most 100

Try It 2.154

Nine times z is no less than 135

Example 2.78

Thirty less than x is at least 45.

Try It 2.155

Nineteen less than p is no less than 47

Try It 2.156

Four more than a is at most 15.

Section 2.7 Exercises

Practice makes perfect.

In the following exercises, graph each inequality on the number line.

ⓐ x ≤ − 2 x ≤ − 2 ⓑ x > − 1 x > − 1 ⓒ x < 0 x < 0

ⓐ x > 1 x > 1 ⓑ x < − 2 x < − 2 ⓒ x ≥ −3 x ≥ −3

ⓐ x ≥ −3 x ≥ −3 ⓑ x < 4 x < 4 ⓒ x ≤ − 2 x ≤ − 2

ⓐ x ≤ 0 x ≤ 0 ⓑ x > − 4 x > − 4 ⓒ x ≥ −1 x ≥ −1

In the following exercises, graph each inequality on the number line and write in interval notation.

ⓐ x < − 2 x < − 2 ⓑ x ≥ −3.5 x ≥ −3.5 ⓒ x ≤ 2 3 x ≤ 2 3

ⓐ x > 3 x > 3 ⓑ x ≤ − 0.5 x ≤ − 0.5 ⓒ x ≥ 1 3 x ≥ 1 3

ⓐ x ≥ −4 x ≥ −4 ⓑ x < 2.5 x < 2.5 ⓒ x > − 3 2 x > − 3 2

ⓐ x ≤ 5 x ≤ 5 ⓑ x ≥ −1.5 x ≥ −1.5 ⓒ x < − 7 3 x < − 7 3

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

n − 11 < 33 n − 11 < 33

m − 45 ≤ 62 m − 45 ≤ 62

u + 25 > 21 u + 25 > 21

v + 12 > 3 v + 12 > 3

a + 3 4 ≥ 7 10 a + 3 4 ≥ 7 10

b + 7 8 ≥ 1 6 b + 7 8 ≥ 1 6

f − 13 20 < − 5 12 f − 13 20 < − 5 12

g − 11 12 < − 5 18 g − 11 12 < − 5 18

8 x > 72 8 x > 72

6 y < 48 6 y < 48

7 r ≤ 56 7 r ≤ 56

9 s ≥ 81 9 s ≥ 81

−5 u ≥ 65 −5 u ≥ 65

−8 v ≤ 96 −8 v ≤ 96

−9 c < 126 −9 c < 126

−7 d > 105 −7 d > 105

20 > 2 5 h 20 > 2 5 h

40 < 5 8 k 40 < 5 8 k

7 6 j ≥ 42 7 6 j ≥ 42

9 4 g ≤ 36 9 4 g ≤ 36

a −3 ≤ 9 a −3 ≤ 9

b −10 ≥ 30 b −10 ≥ 30

−25 < p −5 −25 < p −5

−18 > q −6 −18 > q −6

9 t ≥ −27 9 t ≥ −27

7 s < − 28 7 s < − 28

2 3 y > − 36 2 3 y > − 36

3 5 x ≤ − 45 3 5 x ≤ − 45

4 v ≥ 9 v − 40 4 v ≥ 9 v − 40

5 u ≤ 8 u − 21 5 u ≤ 8 u − 21

13 q < 7 q − 29 13 q < 7 q − 29

9 p > 14 p − 18 9 p > 14 p − 18

12 x + 3 ( x + 7 ) > 10 x − 24 12 x + 3 ( x + 7 ) > 10 x − 24

9 y + 5 ( y + 3 ) < 4 y − 35 9 y + 5 ( y + 3 ) < 4 y − 35

6 h − 4 ( h − 1 ) ≤ 7 h − 11 6 h − 4 ( h − 1 ) ≤ 7 h − 11

4 k − ( k − 2 ) ≥ 7 k − 26 4 k − ( k − 2 ) ≥ 7 k − 26

8 m − 2 ( 14 − m ) ≥ ​ 7 ( m − 4 ) + 3 m 8 m − 2 ( 14 − m ) ≥ ​ 7 ( m − 4 ) + 3 m

6 n − 12 ( 3 − n ) ≤ 9 ( n − 4 ) + 9 n 6 n − 12 ( 3 − n ) ≤ 9 ( n − 4 ) + 9 n

3 4 b − 1 3 b < 5 12 b − 1 2 3 4 b − 1 3 b < 5 12 b − 1 2

9 u + 5 ( 2 u − 5 ) ≥ 12 ( u − 1 ) + 7 u 9 u + 5 ( 2 u − 5 ) ≥ 12 ( u − 1 ) + 7 u

2 3 g − 1 2 ( g − 14 ) ≤ 1 6 ( g + 42 ) 2 3 g − 1 2 ( g − 14 ) ≤ 1 6 ( g + 42 )

5 6 a − 1 4 a > 7 12 a + 2 3 5 6 a − 1 4 a > 7 12 a + 2 3

4 5 h − 2 3 ( h − 9 ) ≥ 1 15 ( 2 h + 90 ) 4 5 h − 2 3 ( h − 9 ) ≥ 1 15 ( 2 h + 90 )

12 v + 3 ( 4 v − 1 ) ≤ 19 ( v − 2 ) + 5 v 12 v + 3 ( 4 v − 1 ) ≤ 19 ( v − 2 ) + 5 v

Mixed practice

15 k ≤ − 40 15 k ≤ − 40

35 k ≥ −77 35 k ≥ −77

23 p − 2 ( 6 − 5 p ) > 3 ( 11 p − 4 ) 23 p − 2 ( 6 − 5 p ) > 3 ( 11 p − 4 )

18 q − 4 ( 10 − 3 q ) < 5 ( 6 q − 8 ) 18 q − 4 ( 10 − 3 q ) < 5 ( 6 q − 8 )

− 9 4 x ≥ − 5 12 − 9 4 x ≥ − 5 12

− 21 8 y ≤ − 15 28 − 21 8 y ≤ − 15 28

c + 34 < − 99 c + 34 < − 99

d + 29 > − 61 d + 29 > − 61

m 18 ≥ −4 m 18 ≥ −4

n 13 ≤ − 6 n 13 ≤ − 6

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 −40 −40 .

Ten times y is at most −110 −110 .

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 −22 −22 .

Fifteen less than a is at least −7 −7 .

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 check-out 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’.

Explain why it is necessary to reverse the inequality when solving −5 x > 10 −5 x > 10 .

Explain why it is necessary to reverse the inequality when solving n −3 < 12 n −3 < 12 .

ⓐ 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?

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• Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Elementary Algebra 2e
• Publication date: Apr 22, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/elementary-algebra-2e/pages/2-7-solve-linear-inequalities

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