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Intermediate Algebra

9.8 Solve Quadratic Inequalities

Intermediate Algebra9.8 Solve Quadratic 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 quadratic inequalities graphically
  • Solve quadratic inequalities algebraically
Be Prepared 9.8

Before you get started, take this readiness quiz.

  1. Solve: 2x3=0.2x3=0.
    If you missed this problem, review Example 2.2.
  2. Solve: 2y2+y=152y2+y=15.
    If you missed this problem, review Example 6.45.
  3. Solve 1x2+2x8>01x2+2x8>0
    If you missed this problem, review Example 7.56.

We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.

We will now learn to solve inequalities that have a quadratic expression. We will use some of the techniques from solving linear and rational inequalities as well as quadratic equations.

We will solve quadratic inequalities two ways—both graphically and algebraically.

Solve Quadratic Inequalities Graphically

A quadratic equation is in standard form when written as ax2 + bx + c = 0. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard form.

Quadratic Inequality

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

The standard form of a quadratic inequality is written:

ax2+bx+c<0ax2+bx+c0 ax2+bx+c>0ax2+bx+c0ax2+bx+c<0ax2+bx+c0 ax2+bx+c>0ax2+bx+c0

The graph of a quadratic function f(x) = ax2 + bx + c = 0 is a parabola. When we ask when is ax2 + bx + c < 0, we are asking when is f(x) < 0. We want to know when the parabola is below the x-axis.

When we ask when is ax2 + bx + c > 0, we are asking when is f(x) > 0. We want to know when the parabola is above the y-axis.

The first graph is an upward facing parabola, f of x, on an x y-coordinate plane. To the left of the function, f of x is greater than 0. Between the x-intercepts, f of x is less than 0. To the right of the function, f of x is greater than 0. The second graph is a downward-facing parabola, f of x, on an x y coordinate plane. To the left of the function, f of x is less than 0. Between the x-intercepts, f of x is greater than 0. To the right of the function, f of x is less than 0.

Example 9.64

How to Solve a Quadratic Inequality Graphically

Solve x26x+8<0x26x+8<0 graphically. Write the solution in interval notation.

Try It 9.127

Solve x2+2x8<0x2+2x8<0 graphically and write the solution in interval notation.

Try It 9.128

Solve x28x+120x28x+120 graphically and write the solution in interval notation.

We list the steps to take to solve a quadratic inequality graphically.

How To

Solve a quadratic inequality graphically.

  1. Step 1. Write the quadratic inequality in standard form.
  2. Step 2. Graph the function f(x)=ax2+bx+c.f(x)=ax2+bx+c.
  3. Step 3. Determine the solution from the graph.

In the last example, the parabola opened upward and in the next example, it opens downward. In both cases, we are looking for the part of the parabola that is below the x-axis but note how the position of the parabola affects the solution.

Example 9.65

Solve x28x120x28x120 graphically. Write the solution in interval notation.

Try It 9.129

Solve x26x5>0x26x5>0 graphically and write the solution in interval notation.

Try It 9.130

Solve x2+10x160x2+10x160 graphically and write the solution in interval notation.

Solve Quadratic Inequalities Algebraically

The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.

We will use the critical points to divide the number line into intervals and then determine whether the quadratic expression willl be postive or negative in the interval. We then determine the solution for the inequality.

Example 9.66

How To Solve Quadratic Inequalities Algebraically

Solve x2x120x2x120 algebraically. Write the solution in interval notation.

Try It 9.131

Solve x2+2x80x2+2x80 algebraically. Write the solution in interval notation.

Try It 9.132

Solve x22x150x22x150 algebraically. Write the solution in interval notation.

In this example, since the expression x2x12x2x12 factors nicely, we can also find the sign in each interval much like we did when we solved rational inequalities. We find the sign of each of the factors, and then the sign of the product. Our number line would like this:

The figure shows the expression x squared minus x minus 12 factored to the quantity of x plus 3 times the quantity of x minus 4. The image shows a number line showing dotted lines on negative 3 and 4. It shows the signs of the quantity x plus 3 to be negative, positive, positive, and the signs of the quantity x minus 4 to be negative, negative, positive. Under the number line, it shows the quantity x plus 3 times the quantity x minus 4 with the signs positive, negative, positive.

The result is the same as we found using the other method.

We summarize the steps here.

How To

Solve a quadratic inequality algebraically.

  1. Step 1. Write the quadratic inequality in standard form.
  2. Step 2. Determine the critical points—the solutions to the related quadratic equation.
  3. Step 3. Use the critical points to divide the number line into intervals.
  4. Step 4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.
  5. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

Example 9.67

Solve x2+6x70x2+6x70 algebraically. Write the solution in interval notation.

Try It 9.133

Solve x2+2x+10x2+2x+10 algebraically. Write the solution in interval notation.

Try It 9.134

Solve x2+8x14<0x2+8x14<0 algebraically. Write the solution in interval notation.

The solutions of the quadratic inequalities in each of the previous examples, were either an interval or the union of two intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation ax2 + bx + c = 0. These two solutions then gave us either the two x-intercepts for the graph or the two critical points to divide the number line into intervals.

This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.

For a quadratic equation of the form ax2 + bx + c = 0, a0.a0.

The figure is a table with 3 columns. Column 1 is labeled discriminant, column 2 is Number/Type of solution, and column 3 is Typical Graph. Reading across the columns, if b squared minus 4 times a times c is greater than 0, there will be 2 real solutions because there are 2 x-intercepts on the graph. The image of a typical graph an upward or downward parabola with 2 x-intercepts. If the discriminant b squared minus 4 times a times c is equals to 0, then there is 1 real solution because there is 1 x-intercept on the graph. The image of the typical graph is an upward- or downward-facing parabola that has a vertex on the x-axis instead of crossing through it. If the discriminant b squared minus 4 times a times c is less than 0, there are 2 complex solutions because there is no x-intercept. The image of the typical graph shows an upward- or downward-facing parabola that does not cross the x-axis.

The last row of the table shows us when the parabolas never intersect the x-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.

In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.

Example 9.68

Solve, writing any solution in interval notation:

x23x+4>0x23x+4>0 x23x+40x23x+40

Try It 9.135

Solve and write any solution in interval notation:
x2+2x40x2+2x40 x2+2x40x2+2x40

Try It 9.136

Solve and write any solution in interval notation:
x2+3x+3<0x2+3x+3<0 x2+3x+3>0x2+3x+3>0

Section 9.8 Exercises

Practice Makes Perfect

Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

363.

x2+6x+5>0x2+6x+5>0

364.

x2+4x12<0x2+4x12<0

365.

x2+4x+30x2+4x+30

366.

x26x+80x26x+80

367.

x23x+180x23x+180

368.

x2+2x+24<0x2+2x+24<0

369.

x2+x+120x2+x+120

370.

x2+2x+15>0x2+2x+15>0

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

371.

x2+3x40x2+3x40

372.

x2+x60x2+x60

373.

x27x+10<0x27x+10<0

374.

x24x+3>0x24x+3>0

375.

x2+8x>15x2+8x>15

376.

x2+8x<12x2+8x<12

377.

x24x+20x24x+20

378.

x2+8x11<0x2+8x11<0

379.

x210x>19x210x>19

380.

x2+6x<3x2+6x<3

381.

−6x2+19x100−6x2+19x100

382.

−3x24x+40−3x24x+40

383.

−2x2+7x+40−2x2+7x+40

384.

2x2+5x12>02x2+5x12>0

385.

x2+3x+5>0x2+3x+5>0

386.

x23x+60x23x+60

387.

x2+x7>0x2+x7>0

388.

x24x5<0x24x5<0

389.

−2x2+8x10<0−2x2+8x10<0

390.

x2+2x70x2+2x70

Writing Exercises

391.

Explain critical points and how they are used to solve quadratic inequalities algebraically.

392.

Solve x2+2x8x2+2x8 both graphically and algebraically. Which method do you prefer, and why?

393.

Describe the steps needed to solve a quadratic inequality graphically.

394.

Describe the steps needed to solve a quadratic inequality algebraically.

Self Check

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

This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can…, Confidently, With some help, and No-I don’t get it! Below I can…, there is solve quadratic inequalities graphically and solve quadratic inequalities algebraically. The other columns are left blank for you to check you understanding.

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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