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Mathematics LibreTexts

19.5: Solve Quadratic Inequalities

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Learning Objectives

By the end of this section, you will be able to:

  • Solve quadratic inequalities graphically
  • Solve quadratic inequalities algebraically

Before you get started, take this readiness quiz.

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

Definition 19.5.1: 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+c0ax2+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.
Figure 9.8.1
Example 19.5.1: How to Solve a Quadratic Inequality Graphically

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

Solution:

Step 1: Write the quadratic inequality in standard form.

The inequality is in standard form.

x26x+8<0

Step 2: Graph the function f(x)=ax2+bx+c using properties or transformations.

We will graph using the properties.

f(x)=x26x+8

Look at a in the equation.

a=1,b=6,c=8

f(x)=x26x+8

Since a is positive, the parabola opens upward.

The parabola opens upward.

Screenshot (2).png
Figure 9.8.2

f(x)=x26x+8

The axis of symmetry is the line x=b2a.

Axis of Symmetry

x=b2a

x=(6)21x=3

The axis of symmetry is the line x=3.

The vertex is on the axis of symmetry. Substitute x=3 into the function.

Vertex

f(x)=x26x+8f(3)=(3)26(3)+8f(3)=1

The vertex is (3,1).

We find f(0)

y-intercept

f(x)=x26x+8f(0)=(0)26(0)+8f(0)=8

The y-intercept is (0,8).

We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x=3. A point 3 units to the right of the axis of symmetry has x=6.

Point symmetric to y-intercept

The point is (6,8).

We solve f(x)=0.

x-intercepts

We can solve this quadratic equation by factoring.

f(x)=x26x+80=x26x+80=(x2)(x4)x=2 or x=4

The x-intercepts are (2,0) and (4,0).

We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola.

Screenshot (3).png
Figure 9.8.3

Step 3: Determine the solution from the graph.

x26x+8<0

The inequality asks for the values of x which make the function less than 0. Which values of x make the parabola below the x-axis.

We do not include the values 2, 4 as the inequality is less than only.

The solution, in interval notation, is (2,4).

Exercise 19.5.1
  1. Solve x2+2x8<0 graphically
  2. Write the solution in interval notation
Answer

  1. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 9), y-intercept of (0, 8), and axis of symmetry shown at x equals negative 2.
    Figure 9.8.4
  2. (4,2)
Exercise 19.5.2
  1. Solve x28x+120 graphically
  2. Write the solution in interval notation
Answer

  1. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and x-intercepts of (2, 0) and (6, 0).
    Figure 9.8.5
  2. (,2][6,)

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

Solve a Quadratic Inequality Graphically

  1. Write the quadratic inequality in standard form.
  2. Graph the function f(x)=ax2+bx+c.
  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 19.5.2

Solve x28x120 graphically. Write the solution in interval notation.

Solution:

The quadratic inequality in standard form. x28x120

Graph the function

f(x)=x28x12

The parabola opens downward.

.
Figure 9.8.6
Find the line of symmetry. x=b2ax=82(1)x=4
Find the vertex.

f(x)=x28x12f(4)=(4)28(4)12f(4)=16+3212f(4)=4

Vertex (4,4)

Find the x-intercepts. Let f(x)=0. f(x)=x28x120=x28x12
Factor: Use the Zero Product Property. 0=1(x+6)(x+2)x=6x=2
Graph the parabola.

x-intercepts (6,0),(2.0)

.
Figure 9.8.7
Determine the solution from the graph. We include the x-intercepts as the inequality is "less than or equal to." (,6][2,)
Table 9.8.1
Exercise 19.5.3
  1. Solve x26x5>0 graphically
  2. Write the solution in interval notation
Answer

  1. A downward-facing parabola on the x y-coordinate plane. It has a vertex of (negative 3, 4), a y-intercept at (0, negative 5), and an axis of symmetry shown at x equals negative 3.
    Figure 9.8.8
  2. (5,1)
Exercise 19.5.4
  1. Solve x2+10x160 graphically
  2. Write the solution in interval notation
Answer

  1. A downward-facing parabola on the x y-coordinate plane. It has a vertex of (5, 9), a y-intercept at (0, negative 16), and an axis of symmetry of x equals 5.
    Figure 9.8.9
  2. (,2][8,)

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 will be positive or negative in the interval. We then determine the solution for the inequality.

Example 19.5.3: How to Solve Quadratic Inequalities Algebraically

Solve x2x120 algebraically. Write the solution in interval notation.

Solution:

Step 1: Write the quadratic inequality in standard form. The inequality is in standard form. x2x120
Step 2: Determine the critical points--the solutions to the related quadratic equation. Change the inequality sign to an equal sign and then solve the equation. x2x12=0(x+3)(x4)=0x+3=0x4=0x=3x=4
Step 3: Use the critical points to divide the number line into intervals. Use 3 and 4 to divide the number line into intervals. Screenshot (4).png
Step 4: Above the number line show the sign of each quadratic expression using test points from each interval substituted from the original inequality.

Test:

x=5

x=0

x=5

x2x12x2x12x2x12(5)2(5)12020125251218128

Screenshot (5).png
Figure 9.8.11
Step 5: Determine the intervals where the inequality is correct. Write the solution in interval notation.

x2x120

The inequality is positive in the first and last intervals and equals 0 at the points 4,3.

The solution, in interval notation, is (,3][4,).
Table 9.8.2
Exercise 19.5.5

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

Answer

(,4][2,)

Exercise 19.5.6

Solve x22x150 algebraically. Write the solution in interval notation.

Answer

[3,5]

In this example, since the expression x2x12 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.
Figure 9.8.12

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

We summarize the steps here.

Solve a Quadratic Inequality Algebraically

  1. Write the quadratic inequality in standard form.
  2. Determine the critical points—the solutions to the related quadratic equation.
  3. Use the critical points to divide the number line into intervals.
  4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.
  5. Determine the intervals where the inequality is correct. Write the solution in interval notation.
Example 19.5.4

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

Solution:

Write the quadratic inequality in standard form. x2+6x70
Multiply both sides of the inequality by 1. Remember to reverse the inequality sign. x26x+70
Determine the critical points by solving the related quadratic equation. x26x+7=0
Write the Quadratic Formula. x=b±b24ac2a
Then substitute in the values of a,b,c. x=(6)±(6)241(7)21
Simplify. x=6±82
Simplify the radical. x=6±222
Remove the common factor, 2. x=2(3±2)2x=3±2x=3+2x=32x1.6x4.4
Use the critical points to divide the number line into intervals. Test numbers from each interval in the original inequality. .
Determine the intervals where the inequality is correct. Write the solution in interval notation. x2+6x70 in the middle interval [32,3+2]
Table 9.8.3
Exercise 19.5.7

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

Answer

[12,1+2]

Exercise 19.5.8

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

Answer

(,42)(4+2,)

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+bc+c=0,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.
Figure 9.8.14

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 19.5.5

Solve, writing any solution in interval notation:

  1. x23x+4>0
  2. x23x+40

Solution:

a.

Write the quadratic inequality in standard form. x23x+4>0
Determine the critical points by solving the related quadratic equation. x23x+4=0
Write the Quadratic Formula. x=b±b24ac2a
Then substitute in the values of a,b,c. x=(3)±(3)241(4)21
Simplify. x=3±72
Simplify the radicand. x=3±7i2
The complex solutions tell us the
parabola does not intercept the x-axis.
Also, the parabola opens upward. This
tells us that the parabola is completely above the x-axis.

Complex solutions

.
Figure 9.8.15
Table 9.8.4

We are to find the solution to x23x+4>0. Since for all values of x the graph is above the x-axis, all values of x make the inequality true. In interval notation we write (,).

b. Write the quadratic inequality in standard form.

x23x+40

Determine the critical points by solving the related quadratic equation.

x23x+4=0

Since the corresponding quadratic equation is the same as in part (a), the parabola will be the same. The parabola opens upward and is completely above the x-axis—no part of it is below the x-axis.

We are to find the solution to x23x+40. Since for all values of x the graph is never below the x-axis, no values of x make the inequality true. There is no solution to the inequality.

Exercise 19.5.9

Solve and write any solution in interval notation:

  1. x2+2x40
  2. x2+2x40
Answer
  1. (,)
  2. no solution
Exercise 19.5.10

Solve and write any solution in interval notation:

  1. x2+3x+3<0
  2. x2+3x+3>0
Answer
  1. no solution
  2. (,)

Key Concepts

  • Solve a Quadratic Inequality Graphically
    1. Write the quadratic inequality in standard form.
    2. Graph the function f(x)=ax2+bx+c using properties or transformations.
    3. Determine the solution from the graph.
  • How to Solve a Quadratic Inequality Algebraically
    1. Write the quadratic inequality in standard form.
    2. Determine the critical points -- the solutions to the related quadratic equation.
    3. Use the critical points to divide the number line into intervals.
    4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.
    5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

Glossary

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

19.5: Solve Quadratic Inequalities is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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