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Intermediate Algebra 2e

2.6 Solve Compound Inequalities

Intermediate Algebra 2e2.6 Solve Compound Inequalities
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Solve compound inequalities with “and”
  • Solve compound inequalities with “or”
  • Solve applications with compound inequalities
Be Prepared 2.15

Before you get started, take this readiness quiz.

Simplify: 25(x+10).25(x+10).
If you missed this problem, review Example 1.51.

Be Prepared 2.16

Simplify: (x4).(x4).
If you missed this problem, review Example 1.54.

Solve Compound Inequalities with “and”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.

x+3>−4and4x53x+3>−4and4x53
2(y+1)<0ory5−22(y+1)<0ory5−2

Compound Inequality

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.

To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true.

Let’s start with the compound inequalities with “and.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streets—the part where the streets overlap—belongs to both streets.

The figure is an illustration of two streets with their intersection shaded

To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap.

For the compound inequality x>−3x>−3 and x2,x2, we graph each inequality. We then look for where the graphs “overlap”. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See Figure 2.5.

The figure shows the graph of x is greater than negative 3 with a left parenthesis at negative 3 and shading to its right, the graph of x is less than or equal to 2 with a bracket at 2 and shading to its left, and the graph of x is greater than negative 3 and x is less than or equal to 2 with a left parenthesis at negative 3 and a right parenthesis at 2 and shading between negative 3 and 2. Negative 3 and 2 are marked by lines on each number line.
Figure 2.5

We can see that the numbers between −3−3 and 22 are shaded on both of the first two graphs. They will then be shaded on the solution graph.

The number −3−3 is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.

The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.

This is how we will show our solution in the next examples.

Example 2.61

Solve 6x3<96x3<9 and 2x+93.2x+93. Graph the solution and write the solution in interval notation.

Try It 2.121

Solve the compound inequality. Graph the solution and write the solution in interval notation: 4x7<94x7<9 and 5x+83.5x+83.

Try It 2.122

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3x4<53x4<5 and 4x+91.4x+91.

How To

Solve a compound inequality with “and.”

  1. Step 1. Solve each inequality.
  2. Step 2. Graph each solution. Then graph the numbers that make both inequalities true.
    This graph shows the solution to the compound inequality.
  3. Step 3. Write the solution in interval notation.

Example 2.62

Solve 3(2x+5)183(2x+5)18 and 2(x7)<−6.2(x7)<−6. Graph the solution and write the solution in interval notation.

Try It 2.123

Solve the compound inequality. Graph the solution and write the solution in interval notation: 2(3x+1)202(3x+1)20 and 4(x1)<2.4(x1)<2.

Try It 2.124

Solve the compound inequality. Graph the solution and write the solution in interval notation: 5(3x1)105(3x1)10 and 4(x+3)<8.4(x+3)<8.

Example 2.63

Solve 13x4−213x4−2 and −2(x3)4.−2(x3)4. Graph the solution and write the solution in interval notation.

Try It 2.125

Solve the compound inequality. Graph the solution and write the solution in interval notation: 14x3−114x3−1 and −3(x2)2.−3(x2)2.

Try It 2.126

Solve the compound inequality. Graph the solution and write the solution in interval notation: 15x5−315x5−3 and −4(x1)−2.−4(x1)−2.

Sometimes we have a compound inequality that can be written more concisely. For example, a<xa<x and x<bx<b can be written simply as a<x<ba<x<b and then we call it a double inequality. The two forms are equivalent.

Double Inequality

A double inequality is a compound inequality such as a<x<b.a<x<b. It is equivalent to a<xa<x and x<b.x<b.

Other forms:a<x<bis equivalent toa<xandx<baxbis equivalent toaxandxba>x>bis equivalent toa>xandx>baxbis equivalent toaxandxbOther forms:a<x<bis equivalent toa<xandx<baxbis equivalent toaxandxba>x>bis equivalent toa>xandx>baxbis equivalent toaxandxb

To solve a double inequality we perform the same operation on all three “parts” of the double inequality with the goal of isolating the variable in the center.

Example 2.64

Solve −43x7<8.−43x7<8. Graph the solution and write the solution in interval notation.

When written as a double inequality, 1x<5,1x<5, it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.

Another way to graph the solution of 1x<51x<5 is to graph both the solution of x1x1 and the solution of x<5.x<5. We would then find the numbers that make both inequalities true as we did in previous examples.

Try It 2.127

Solve the compound inequality. Graph the solution and write the solution in interval notation: −54x1<7.−54x1<7.

Try It 2.128

Solve the compound inequality. Graph the solution and write the solution in interval notation: −3<2x51.−3<2x51.

Solve Compound Inequalities with “or”

To solve a compound inequality with “or”, we start out just as we did with the compound inequalities with “and”—we solve the two inequalities. Then we find all the numbers that make either inequality true.

Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.

To write the solution in interval notation, we will often use the union symbol, to show the union of the solutions shown in the graphs.

How To

Solve a compound inequality with “or.”

  1. Step 1. Solve each inequality.
  2. Step 2. Graph each solution. Then graph the numbers that make either inequality true.
  3. Step 3. Write the solution in interval notation.

Example 2.65

Solve 53x−153x−1 or 8+2x5.8+2x5. Graph the solution and write the solution in interval notation.

Try It 2.129

Solve the compound inequality. Graph the solution and write the solution in interval notation: 12x−312x−3 or 7+3x4.7+3x4.

Try It 2.130

Solve the compound inequality. Graph the solution and write the solution in interval notation: 25x−325x−3 or 5+2x3.5+2x3.

Example 2.66

Solve 23x4323x43 or 14(x+8)−1.14(x+8)−1. Graph the solution and write the solution in interval notation.

Try It 2.131

Solve the compound inequality. Graph the solution and write the solution in interval notation: 35x7−135x7−1 or 13(x+6)−2.13(x+6)−2.

Try It 2.132

Solve the compound inequality. Graph the solution and write the solution in interval notation: 34x3334x33 or 25(x+10)0.25(x+10)0.

Solve Applications with Compound Inequalities

Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.

Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a compound inequality. Last, we will solve the compound inequality.

Example 2.67

Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $57.06 and $171.02. How many hcf can the owner use if he wants his usage to stay in the normal range?

Try It 2.133

Due to the drought in California, many communities now have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.32 per hcf for Conservation Usage. The bill for Conservation Usage would be between or equal to $31.32 and $52.12. How many hcf can the owner use if she wants her usage to stay in the conservation range?

Try It 2.134

Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the winter, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $49.36 and $86.32. How many hcf will he be allowed to use if he wants his usage to stay in the normal range?

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with solving compound inequalities.

Section 2.6 Exercises

Practice Makes Perfect

Solve Compound Inequalities with “and”

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

376.

x<3x<3 and x1x1

377.

x4x4 and x>−2x>−2

378.

x−4x−4 and x−1x−1

379.

x>−6x>−6 and x<−3x<−3

380.

5x2<85x2<8 and 6x+936x+93

381.

4x1<74x1<7 and 2x+842x+84

382.

4x+624x+62 and
2x+1−52x+1−5

383.

4x244x24 and
7x1>−87x1>−8

384.

2x11<52x11<5 and
3x8>−53x8>−5

385.

7x8<67x8<6 and
5x+7>−35x+7>−3

386.

4(2x1)124(2x1)12 and
2(x+1)<42(x+1)<4

387.

5(3x2)55(3x2)5 and
3(x+3)<33(x+3)<3

388.

3(2x3)>33(2x3)>3 and
4(x+5)44(x+5)4

389.

−3(x+4)<0−3(x+4)<0 and
−1(3x1)7−1(3x1)7

390.

12(3x4)112(3x4)1 and
13(x+6)413(x+6)4

391.

34(x8)334(x8)3 and
15(x5)315(x5)3

392.

5x23x+45x23x+4 and
3x42x+13x42x+1

393.

34x5−234x5−2 and
−3(x+1)6−3(x+1)6

394.

23x6−423x6−4 and
−4(x+2)0−4(x+2)0

395.

12(x6)+2<−512(x6)+2<−5 and
423x<6423x<6

396.

−54x1<7−54x1<7

397.

−3<2x51−3<2x51

398.

5<4x+1<95<4x+1<9

399.

−1<3x+2<8−1<3x+2<8

400.

−8<5x+2−3−8<5x+2−3

401.

−64x2<−2−64x2<−2

Solve Compound Inequalities with “or”

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

402.

x−2x−2 or x>3x>3

403.

x−4x−4 or x>−3x>−3

404.

x<2x<2 or x5x5

405.

x<0x<0 or x4x4

406.

2+3x42+3x4 or
52x−152x−1

407.

43x−243x−2 or
2x1−52x1−5

408.

2(3x1)<42(3x1)<4 or
3x5>13x5>1

409.

3(2x3)<−53(2x3)<−5 or
4x1>34x1>3

410.

34x2>434x2>4 or 4(2x)>04(2x)>0

411.

23x3>523x3>5 or 3(5x)>63(5x)>6

412.

3x2>43x2>4 or 5x375x37

413.

2(x+3)02(x+3)0 or
3(x+4)63(x+4)6

414.

12x3412x34 or
13(x6)−213(x6)−2

415.

34x+2−134x+2−1 or
12(x+8)−312(x+8)−3

Mixed practice

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

416.

3x+713x+71 and
2x+3−52x+3−5

417.

6(2x1)>66(2x1)>6 and
5(x+2)05(x+2)0

418.

47x−347x−3 or
5(x3)+8>35(x3)+8>3

419.

12x5312x53 or
14(x8)−314(x8)−3

420.

−52x1<7−52x1<7

421.

15(x5)+6<415(x5)+6<4 and
323x<5323x<5

422.

4x2>64x2>6 or
3x1−23x1−2

423.

6x316x31 and
5x1>−65x1>−6

424.

−2(3x4)2−2(3x4)2 and
−4(x1)<2−4(x1)<2

425.

−53x24−53x24

Solve Applications with Compound Inequalities

In the following exercises, solve.

426.

Penelope is playing a number game with her sister June. Penelope is thinking of a number and wants June to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Penelope might be thinking of.

427.

Gregory is thinking of a number and he wants his sister Lauren to guess the number. His first clue is that six less than twice his number is between four and forty-two. Write a compound inequality that shows the range of numbers that Gregory might be thinking of.

428.

Andrew is creating a rectangular dog run in his back yard. The length of the dog run is 18 feet. The perimeter of the dog run must be at least 42 feet and no more than 72 feet. Use a compound inequality to find the range of values for the width of the dog run.

429.

Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.

Everyday Math

430.

Blood Pressure A person’s blood pressure is measured with two numbers. The systolic blood pressure measures the pressure of the blood on the arteries as the heart beats. The diastolic blood pressure measures the pressure while the heart is resting.

Let x be your systolic blood pressure. Research and then write the compound inequality that shows you what a normal systolic blood pressure should be for someone your age.

Let y be your diastolic blood pressure. Research and then write the compound inequality that shows you what a normal diastolic blood pressure should be for someone your age.

431.

Body Mass Index (BMI) is a measure of body fat is determined using your height and weight.

Let x be your BMI. Research and then write the compound inequality to show the BMI range for you to be considered normal weight.

Research a BMI calculator and determine your BMI. Is it a solution to the inequality in part (a)?

Writing Exercises

432.

In your own words, explain the difference between the properties of equality and the properties of inequality.

433.

Explain the steps for solving the compound inequality 27x−527x−5 or 4(x3)+7>3.4(x3)+7>3.

Self Check

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

This table has four columns and four rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was solve compound inequalities with “and.” In row 3, the I can was solve compound inequalities with “or.” In row 4, the I can was solve applications with compound inequalities.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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