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Intermediate Algebra

7.6 Solve Rational Inequalities

Intermediate Algebra7.6 Solve Rational 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 rational inequalities
  • Solve an inequality with rational functions
Be Prepared 7.6

Before you get started, take this readiness quiz.

  1. Find the value of x5x5 when x=6x=6 x=−3x=−3 x=5.x=5.
    If you missed this problem, review Example 1.6.
  2. Solve: 82x<12.82x<12.
    If you missed this problem, review Example 2.52.
  3. Write in interval notation: −3x<5.−3x<5.
    If you missed this problem, review Example 2.49.

Solve Rational Inequalities

We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.

Having just learned to solve rational equations we are now ready to solve rational inequalities. A rational inequality is an inequality that contains a rational expression.

Rational Inequality

A rational inequality is an inequality that contains a rational expression.

Inequalities such as 32x>1,2xx3<4,2x3x6x,32x>1,2xx3<4,2x3x6x, and 142x23x142x23x are rational inequalities as they each contain a rational expression.

When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.

Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.

When we solve an equation and the result is x=3,x=3, we know there is one solution, which is 3.

When we solve an inequality and the result is x>3,x>3, we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a critical point and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.

This figure shows the solution, the interval 3 to infinity, of the inequality x is greater than 3 on a number line. The values range from negative 5 to 5 on the number line. The inequality is modeled by an open parenthesis at the critical point 3 and shading the right.

To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.

Next we determine the critical points to use to divide the number line into intervals. A critical point is a number which make the rational expression zero or undefined.

We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.

We write the solution in interval notation being careful to determine whether the endpoints are included.

Example 7.54

Solve and write the solution in interval notation: x1x+30.x1x+30.

Try It 7.107

Solve and write the solution in interval notation: x2x+40.x2x+40.

Try It 7.108

Solve and write the solution in interval notation: x+2x40.x+2x40.

We summarize the steps for easy reference.

How To

Solve a rational inequality.

  1. Step 1. Write the inequality as one quotient on the left and zero on the right.
  2. Step 2. Determine the critical points–the points where the rational expression will be zero or undefined.
  3. Step 3. Use the critical points to divide the number line into intervals.
  4. Step 4. Test a value in each interval. Above the number line show the sign of each factor of the numerator and denominator in each interval. Below the number line show the sign of the quotient.
  5. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

The next example requires that we first get the rational inequality into the correct form.

Example 7.55

Solve and write the solution in interval notation: 4xx6<1.4xx6<1.

Try It 7.109

Solve and write the solution in interval notation: 3xx3<1.3xx3<1.

Try It 7.110

Solve and write the solution in interval notation: 3xx4<2.3xx4<2.

In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.

Example 7.56

Solve and write the solution in interval notation: 5x22x15>0.5x22x15>0.

Try It 7.111

Solve and write the solution in interval notation: 1x2+2x8>0.1x2+2x8>0.

Try It 7.112

Solve and write the solution in interval notation: 3x2+x12>0.3x2+x12>0.

The next example requires some work to get it into the needed form.

Example 7.57

Solve and write the solution in interval notation: 132x2<53x.132x2<53x.

Try It 7.113

Solve and write the solution in interval notation: 12+4x2<3x.12+4x2<3x.

Try It 7.114

Solve and write the solution in interval notation: 13+6x2<3x.13+6x2<3x.

Solve an Inequality with Rational Functions

When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.

Example 7.58

Given the function R(x)=x+3x5,R(x)=x+3x5, find the values of x that make the function less than or equal to 0.

Try It 7.115

Given the function R(x)=x2x+4,R(x)=x2x+4, find the values of x that make the function less than or equal to 0.

Try It 7.116

Given the function R(x)=x+1x4,R(x)=x+1x4, find the values of x that make the function less than or equal to 0.

In economics, the function C(x)C(x) is used to represent the cost of producing x units of a commodity. The average cost per unit can be found by dividing C(x)C(x) by the number of items x.x. Then, the average cost per unit is c(x)=C(x)x.c(x)=C(x)x.

Example 7.59

The function C(x)=10x+3000C(x)=10x+3000 represents the cost to produce x,x, number of items. Find the average cost function, c(x)c(x) how many items should be produced so that the average cost is less than $40.

Try It 7.117

The function C(x)=20x+6000C(x)=20x+6000 represents the cost to produce x,x, number of items. Find the average cost function, c(x)c(x) how many items should be produced so that the average cost is less than $60?

Try It 7.118

The function C(x)=5x+900C(x)=5x+900 represents the cost to produce x,x, number of items. Find the average cost function, c(x)c(x) how many items should be produced so that the average cost is less than $20?

Section 7.6 Exercises

Practice Makes Perfect

Solve Rational Inequalities

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

339.

x3x+40x3x+40

340.

x+6x50x+6x50

341.

x+1x30x+1x30

342.

x4x+20x4x+20

343.

x7x1>0x7x1>0

344.

x+8x+3>0x+8x+3>0

345.

x6x+5<0x6x+5<0

346.

x+5x2<0x+5x2<0

347.

3xx5<13xx5<1

348.

5xx2<15xx2<1

349.

6xx6>26xx6>2

350.

3xx4>23xx4>2

351.

2x+3x612x+3x61

352.

4x1x414x1x41

353.

3x2x423x2x42

354.

4x3x324x3x32

355.

1x2+7x+12>01x2+7x+12>0

356.

1x24x12>01x24x12>0

357.

3x25x+4<03x25x+4<0

358.

4x2+7x+12<04x2+7x+12<0

359.

22x2+x15022x2+x150

360.

63x22x5063x22x50

361.

−26x213x+60−26x213x+60

362.

−110x2+11x60−110x2+11x60

363.

12+12x2>5x12+12x2>5x

364.

13+1x2>43x13+1x2>43x

365.

124x21x124x21x

366.

1232x21x1232x21x

367.

1x216<01x216<0

368.

4x225>04x225>0

369.

4x23x+14x23x+1

370.

5x14x+25x14x+2

Solve an Inequality with Rational Functions

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

371.

Given the function R(x)=x5x2,R(x)=x5x2, find the values of xx that make the function less than or equal to 0.

372.

Given the function R(x)=x+1x+3,R(x)=x+1x+3, find the values of xx that make the function less than or equal to 0.

373.

Given the function R(x)=x6x+2R(x)=x6x+2, find the values of x that make the function less than or equal to 0.

374.

Given the function R(x)=x+1x4,R(x)=x+1x4, find the values of x that make the function less than or equal to 0.

Writing Exercises

375.

Write the steps you would use to explain solving rational inequalities to your little brother.

376.

Create a rational inequality whose solution is (,−2][4,).(,−2][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 three 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 rational inequalities. In row 3, the I can was solve an inequality with rational functions.

After reviewing this checklist, what will you do to become confident for all objectives?

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