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Intermediate Algebra 2e

2.7 Solve Absolute Value Inequalities

Intermediate Algebra 2e2.7 Solve Absolute Value 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 absolute value equations
  • Solve absolute value inequalities with “less than”
  • Solve absolute value inequalities with “greater than”
  • Solve applications with absolute value
Be Prepared 2.17

Before you get started, take this readiness quiz.

Evaluate: |7|.|7|.
If you missed this problem, review Example 1.12.

Be Prepared 2.18

Fill in <,>,<,>, or == for each of the following pairs of numbers.
|−8|___|−8||−8|___|−8| 12___|−12|12___|−12| |−6|___6|−6|___6 (−15)___|−15|(−15)___|−15|
If you missed this problem, review Example 1.12.

Be Prepared 2.19

Simplify: 142|83(41)|.142|83(41)|.
If you missed this problem, review Example 1.13.

Solve Absolute Value Equations

As we prepare to solve absolute value equations, we review our definition of absolute value.

Absolute Value

The absolute value of a number is its distance from zero on the number line.

The absolute value of a number n is written as |n||n| and |n|0|n|0 for all numbers.

Absolute values are always greater than or equal to zero.

We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:

−5−5 is 5 units away from 0, so |−5|=5.|−5|=5.

55 is 5 units away from 0, so |5|=5.|5|=5.

Figure 2.6 illustrates this idea.

The figure is a number line with tick marks at negative 5, 0, and 5. The distance between negative 5 and 0 is given as 5 units, so the absolute value of negative 5 is 5. The distance between 5 and 0 is 5 units, so the absolute value of 5 is 5.
Figure 2.6 The numbers 5 and −5−5 are both five units away from zero.

For the equation |x|=5,|x|=5, we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and −5−5 are five units from zero on the number line. They are the solutions to the equation.

If|x|=5thenx=−5orx=5If|x|=5thenx=−5orx=5

The solution can be simplified to a single statement by writing x=±5.x=±5. This is read, “x is equal to positive or negative 5”.

We can generalize this to the following property for absolute value equations.

Absolute Value Equations

For any algebraic expression, u, and any positive real number, a,

if|u|=athenu=aoru=aif|u|=athenu=aoru=a

Remember that an absolute value cannot be a negative number.

Example 2.68

Solve: |x|=8|x|=8 |y|=−6|y|=−6 |z|=0|z|=0

Try It 2.135

Solve: |x|=2|x|=2 |y|=−4|y|=−4 |z|=0|z|=0

Try It 2.136

Solve: |x|=11|x|=11 |y|=−5|y|=−5 |z|=0|z|=0

To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.

Example 2.69

How to Solve Absolute Value Equations

Solve |5x4|3=8.|5x4|3=8.

Try It 2.137

Solve: |3x5|1=6.|3x5|1=6.

Try It 2.138

Solve: |4x3|5=2.|4x3|5=2.

The steps for solving an absolute value equation are summarized here.

How To

Solve absolute value equations.

  1. Step 1. Isolate the absolute value expression.
  2. Step 2. Write the equivalent equations.
  3. Step 3. Solve each equation.
  4. Step 4. Check each solution.

Example 2.70

Solve 2|x7|+5=9.2|x7|+5=9.

Try It 2.139

Solve: 3|x4|4=8.3|x4|4=8.

Try It 2.140

Solve: 2|x5|+3=9.2|x5|+3=9.

Remember, an absolute value is always positive!

Example 2.71

Solve: |23x4|+11=3.|23x4|+11=3.

Try It 2.141

Solve: |34x5|+9=4.|34x5|+9=4.

Try It 2.142

Solve: |56x+3|+8=6.|56x+3|+8=6.

Some of our absolute value equations could be of the form |u|=|v||u|=|v| where u and v are algebraic expressions. For example, |x3|=|2x+1|.|x3|=|2x+1|.

How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if |u|=a,|u|=a, then u=au=a or u=a.u=a.

This tell us that

if|u|=|v|thenu=voru=voru=vif|u|=|v|thenu=voru=voru=v

This leads us to the following property for equations with two absolute values.

Equations with Two Absolute Values

For any algebraic expressions, u and v,

if|u|=|v|thenu=voru=vif|u|=|v|thenu=voru=v

When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.

Example 2.72

Solve: |5x1|=|2x+3|.|5x1|=|2x+3|.

Try It 2.143

Solve: |7x3|=|3x+7|.|7x3|=|3x+7|.

Try It 2.144

Solve: |6x5|=|3x+4|.|6x5|=|3x+4|.

Solve Absolute Value Inequalities with “Less Than”

Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.

Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation |x|=5,|x|=5, we saw that both 5 and −5−5 are five units from zero on the number line. They are the solutions to the equation.

|x|=5x=−5orx=5|x|=5x=−5orx=5

What about the inequality |x|5?|x|5? Where are the numbers whose distance is less than or equal to 5? We know −5−5 and 5 are both five units from zero. All the numbers between −5−5 and 5 are less than five units from zero. See Figure 2.7.

The figure is a number line with negative 5, 0, and 5 displayed. There is a left bracket at negative 5 and a right bracket at 5. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x which is less than or equal to 5.
Figure 2.7

In a more general way, we can see that if |u|a,|u|a, then aua.aua. See Figure 2.8.

The figure is a number line with negative a 0, and a displayed. There is a left bracket at negative a and a right bracket at a. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is less than or equal to a, then negative a is less than or equal to u which is less than or equal to a.
Figure 2.8

This result is summarized here.

Absolute Value Inequalities with << or

For any algebraic expression, u, and any positive real number, a,

if|u|<a,thena<u<aif|u|a,thenauaif|u|<a,thena<u<aif|u|a,thenaua

After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.

Example 2.73

Solve |x|<7.|x|<7. Graph the solution and write the solution in interval notation.

Try It 2.145

Graph the solution and write the solution in interval notation: |x|<9.|x|<9.

Try It 2.146

Graph the solution and write the solution in interval notation: |x|<1.|x|<1.

Example 2.74

Solve |5x6|4.|5x6|4. Graph the solution and write the solution in interval notation.

Try It 2.147

Solve |2x1|5.|2x1|5. Graph the solution and write the solution in interval notation:

Try It 2.148

Solve |4x5|3.|4x5|3. Graph the solution and write the solution in interval notation:

How To

Solve absolute value inequalities with < or ≤.

  1. Step 1. Isolate the absolute value expression.
  2. Step 2. Write the equivalent compound inequality.
    |u|<ais equivalent toa<u<a|u|ais equivalent toaua|u|<ais equivalent toa<u<a|u|ais equivalent toaua
  3. Step 3. Solve the compound inequality.
  4. Step 4. Graph the solution
  5. Step 5. Write the solution using interval notation.

Solve Absolute Value Inequalities with “Greater Than”

What happens for absolute value inequalities that have “greater than”? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.

We started with the inequality |x|5.|x|5. We saw that the numbers whose distance is less than or equal to five from zero on the number line were −5−5 and 5 and all the numbers between −5−5 and 5. See Figure 2.9.

The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its right and a right bracket at 5 with shading to its left. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x is less than or equal to 5.
Figure 2.9

Now we want to look at the inequality |x|5.|x|5. Where are the numbers whose distance from zero is greater than or equal to five?

Again both −5−5 and 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than −5−5 and greater than 5 on the number line. See Figure 2.10.

The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its left and a left bracket at 5 with shading to its right. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is greater than or equal to 5, then x is less than or equal to negative 5 or x is greater than or equal to 5.
Figure 2.10

In a more general way, we can see that if |u|a,|u|a, then uaua or ua.ua. See Figure 2.11.

The figure is a number line with negative a, 0, and a displayed. There is a right bracket at negative a that has shading to its left and a left bracket at a with shading to its right. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is greater than or equal to a, then u is less than or equal to negative a or u is greater than or equal to a.
Figure 2.11

This result is summarized here.

Absolute Value Inequalities with > or ≥

For any algebraic expression, u, and any positive real number, a,

if|u|>a,thenu<aoru>aif|u|a,thenuaoruaif|u|>a,thenu<aoru>aif|u|a,thenuaorua

Example 2.75

Solve |x|>4.|x|>4. Graph the solution and write the solution in interval notation.

Try It 2.149

Solve |x|>2.|x|>2. Graph the solution and write the solution in interval notation.

Try It 2.150

Solve |x|>1.|x|>1. Graph the solution and write the solution in interval notation.

Example 2.76

Solve |2x3|5.|2x3|5. Graph the solution and write the solution in interval notation.

Try It 2.151

Solve |4x3|5.|4x3|5. Graph the solution and write the solution in interval notation.

Try It 2.152

Solve |3x4|2.|3x4|2. Graph the solution and write the solution in interval notation.

How To

Solve absolute value inequalities with > or ≥.

  1. Step 1. Isolate the absolute value expression.
  2. Step 2. Write the equivalent compound inequality.
    |u|>ais equivalent tou<aoru>a|u|ais equivalent touaorua|u|>ais equivalent tou<aoru>a|u|ais equivalent touaorua
  3. Step 3. Solve the compound inequality.
  4. Step 4. Graph the solution
  5. Step 5. Write the solution using interval notation.

Solve Applications with Absolute Value

Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain tolerance of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.

|actual-ideal|tolerance|actual-ideal|tolerance

Example 2.77

The ideal diameter of a rod needed for a machine is 60 mm. The actual diameter can vary from the ideal diameter by 0.0750.075 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Try It 2.153

The ideal diameter of a rod needed for a machine is 80 mm. The actual diameter can vary from the ideal diameter by 0.009 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Try It 2.154

The ideal diameter of a rod needed for a machine is 75 mm. The actual diameter can vary from the ideal diameter by 0.05 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with solving linear absolute value equations and inequalities.

Section 2.7 Exercises

Practice Makes Perfect

Solve Absolute Value Equations

In the following exercises, solve.

434.

|x|=6|x|=6 |y|=−3|y|=−3 |z|=0|z|=0

435.

|x|=4|x|=4 |y|=−5|y|=−5 |z|=0|z|=0

436.

|x|=7|x|=7 |y|=−11|y|=−11 |z|=0|z|=0

437.

|x|=3|x|=3 |y|=−1|y|=−1 |z|=0|z|=0

438.

|2x3|4=1|2x3|4=1

439.

|4x1|3=0|4x1|3=0

440.

|3x4|+5=7|3x4|+5=7

441.

|4x+7|+2=5|4x+7|+2=5

442.

4|x1|+2=104|x1|+2=10

443.

3|x4|+2=113|x4|+2=11

444.

3|4x5|4=113|4x5|4=11

445.

3|x+2|5=43|x+2|5=4

446.

−2|x3|+8=−4−2|x3|+8=−4

447.

−3|x4|+4=−5−3|x4|+4=−5

448.

|34x3|+7=2|34x3|+7=2

449.

|35x2|+5=2|35x2|+5=2

450.

|12x+5|+4=1|12x+5|+4=1

451.

|14x+3|+3=1|14x+3|+3=1

452.

|3x2|=|2x3||3x2|=|2x3|

453.

|4x+3|=|2x+1||4x+3|=|2x+1|

454.

|6x5|=|2x+3||6x5|=|2x+3|

455.

|6x|=|32x||6x|=|32x|

Solve Absolute Value Inequalities with “less than”

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

456.

|x|<5|x|<5

457.

|x|<1|x|<1

458.

|x|8|x|8

459.

|x|3|x|3

460.

|3x3|6|3x3|6

461.

|2x5|3|2x5|3

462.

|2x+3|+5<4|2x+3|+5<4

463.

|3x7|+3<1|3x7|+3<1

464.

|4x3|<1|4x3|<1

465.

|6x5|<7|6x5|<7

466.

|x4|−1|x4|−1

467.

|5x+1|−2|5x+1|−2

Solve Absolute Value Inequalities with “greater than”

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

468.

|x|>3|x|>3

469.

|x|>6|x|>6

470.

|x|2|x|2

471.

|x|5|x|5

472.

|3x8|>1|3x8|>1

473.

|x5|>2|x5|>2

474.

|3x2|>4|3x2|>4

475.

|2x1|>5|2x1|>5

476.

|x+3|5|x+3|5

477.

|x7|1|x7|1

478.

3|x|+413|x|+41

479.

5|x|+615|x|+61

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

480.

2|x+6|+4=82|x+6|+4=8

481.

|6x5|=|2x+3||6x5|=|2x+3|

482.

|3x4|2|3x4|2

483.

|2x5|+2=3|2x5|+2=3

484.

|4x3|<5|4x3|<5

485.

|3x+1|3=7|3x+1|3=7

486.

|7x+2|+8<4|7x+2|+8<4

487.

5|2x1|3=75|2x1|3=7

488.

|8x|=|43x||8x|=|43x|

489.

|x7|>3|x7|>3

Solve Applications with Absolute Value

In the following exercises, solve.

490.

A chicken farm ideally produces 200,000 eggs per day. But this total can vary by as much as 25,000 eggs. What is the maximum and minimum expected production at the farm?

491.

An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?

492.

In order to insure compliance with the law, Miguel routinely overshoots the weight of his tortillas by 0.5 gram. He just received a report that told him that he could be losing as much as $100,000 per year using this practice. He now plans to buy new equipment that guarantees the thickness of the tortilla within 0.005 inches. If the ideal thickness of the tortilla is 0.04 inches, what thickness of tortillas will be guaranteed?

493.

At Lilly’s Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?

Writing Exercises

494.

Write a graphical description of the absolute value of a number.

495.

In your own words, explain how to solve the absolute value inequality, |3x2|4.|3x2|4.

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 five 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 absolute value equations. In row 3, the I can was solve absolute value inequalities with “less than.” In row 4, the I can was solve absolute value inequalities with “greater than.” In row 5, the I can was solve applications with absolute value.

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

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