How to Solve Quadratic Inequalities

How to Solve Quadratic Inequalities

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A quadratic inequality has two roots, or x-intercepts because it has an \(x^2\) term. You get a parabola when you plot this inequality on a coordinate plane. Finding the values of \(x\) that make the inequality true is what it means to solve an inequality. By drawing the inequality, you can show these solutions in algebra or on a number line or coordinate plane.

Follow these steps to figure out how to solve a quadratic inequality:

  • Step1:
    Solve the inequality as if it were an equation.
  • Step2:
    The boundary points for the solution to the inequality are the real answers to the equation.
  • Step3:
    If the original inequality did not include equality, make the boundary points open circles. If it did, make the boundary points solid circles.
  • Step4:
    Choose points from each of the areas that the boundary points make. Change these "test points" in the original inequality.
  • Step5:
    If a test point meets the original inequality, then the area that contains that test point is part of the solution.
  • Step6:
    Show the solution both as a picture and as a solution set.

Example:

Solve \((x \ + \ 1) \ (2x \ - \ 4) \ < \ 0\).

Figure out \((x \ + \ 1) \ (2x \ - \ 4) \ = \ 0\). Using the zero-product property,

\((x \ + \ 1) \ = \ 0\) or \((2x \ - \ 4) \ = \ 0\) \(⇒ \ x \ = \ -1 \ , \ x \ = \ 2\)

Make the points of the border. Here, the boundary points are open circles because the original inequality doesn't include equality.

Points of the border:

Quadratic Inequality

There are now three different areas:

Quadratic Inequality2

Choose test points from these three areas: \(x \ = \ -3 \ , \ x \ = \ 0 \ , \ x \ = \ 3\)

Check to see if the test points meet the first inequality.

  • \(x \ = \ -3 \ ⇒ \ (-3 \ + \ 1) \ (2(-3) \ - \ 4) \ < \ 0 \ ⇒ \ (-2)(-10) \ < \ 0 \ ⇒ \ 20 \ < \ 0\)
  • \(x \ = \ 0 \ ⇒ \ (0 \ + \ 1) \ (2(0) \ - \ 4) \ < \ 0 \ ⇒ \ (1)(-4) \ < \ 0 \ ⇒ \ -4 \ < \ 0\)
  • \(x \ = \ 3 \ ⇒ \ (3 \ + \ 1) \ (2(3) \ - \ 4) \ < \ 0 \ ⇒ \ (4)(2) \ < \ 0 \ ⇒ \ 8 \ < \ 0\)

Since \(x \ = \ 0\) solves the original problem, the area \(-1 \ < \ x \ < \ 2\) is part of the answer. Since \(x \ = \ -3\) doesn't solve the original problem, the area \(x \ < \ -1\) isn't part of the answer. Since \(x \ = \ 3\) doesn't solves the original problem, the area \(x \ > \ 2\) isn't part of the answer. Now show the solution in a graphic form and a solution set.

The graphic form:

Quadratic Inequality3

The form of the solution set: \(\{x \ | \ -1 \ < \ x \ < \ 2\}\)

Free printable Worksheets

Exercises for Solving Quadratic Inequalities

1) Solve: \((2x \ + \ 2)(3x \ - \ 6) \ > \ 0\)

2) Solve: \((2x \ + \ 6)(3x \ - \ 12) \ < \ 0\)

3) Solve: \((-2x \ + \ 6)(4x \ + \ 4) \ < \ 0\)

4) Solve: \(2x^2 \ - \ 10x \ + \ 8 \ < \ 0\)

5) Solve: \(3x^2 \ + \ 12x \ + \ 9 \ > \ 0\)

6) Solve: \(x^2 \ - \ 1 \ > \ 0\)

7) Solve: \(x^2 \ - \ 4 \ < \ 0\)

8) Solve: \((2x \ + \ 2)(3x \ - \ 6) \ < \ 0\)

9) Solve: \(x^2 \ - \ x \ - \ 12 \ > \ 0\)

10) Solve: \((-x \ - \ 1)(x \ + \ 3) \ > \ 0\)

 

1) Solve: \((2x \ + \ 2)(3x \ - \ 6) \ > \ 0\)

\(\color{red}{x_1 \ = \ -1, \ x_2 \ = \ 2}\)
Quadratic Inequalities
Test points: \(\color{red}{x \ = \ -2, \ x \ = \ 0, x \ = \ 3}\)
So, \(\color{red}{⇒ \ x \ < \ -1, \ x \ > \ 2}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ x \ < \ -1, \ x \ > \ 2\}}\)

2) Solve: \((2x \ + \ 6)(3x \ - \ 12) \ < \ 0\)

\(\color{red}{x_1 \ = \ -3, \ x_2 \ = \ 4}\)
Quadratic Inequalities2
Test points: \(\color{red}{x \ = \ -4, \ x \ = \ 0, x \ = \ 5}\)
So, \(\color{red}{⇒ \ -3 \ < \ x \ < \ 4}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ -3 \ < \ x \ < \ 4\}}\)

3) Solve: \((-2x \ + \ 6)(4x \ + \ 4) \ < \ 0\)

\(\color{red}{x_1 \ = \ 3, \ x_2 \ = \ -1}\)
Quadratic Inequalities3

Test points: \(\color{red}{x \ = \ -2, \ x \ = \ 0, x \ = \ 4}\)
So, \(\color{red}{⇒ \ x \ < \ -1, \ x \ > \ 3}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ x \ < \ -1, \ x \ > \ 3\}}\)

4) Solve: \(2x^2 \ - \ 10x \ + \ 8 \ < \ 0\)

\(\color{red}{x_1 \ = \ 1, \ x_2 \ = \ 4}\)
Quadratic Inequalities4
Test points: \(\color{red}{x \ = \ 0, \ x \ = \ 2, x \ = \ 5}\)
So, \(\color{red}{⇒ \ 1 \ < \ x \ < \ 4}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ 1 \ < \ x \ < \ 4\}}\)

5) Solve: \(3x^2 \ + \ 12x \ + \ 9 \ > \ 0\)

\(\color{red}{x_1 \ = \ -1, \ x_2 \ = \ -3}\)
Quadratic Inequalities5
Test points: \(\color{red}{x \ = \ -4, \ x \ = \ -2, x \ = \ 0}\)
So, \(\color{red}{⇒ \ x \ < \ -3, \ x \ > \ 4}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ x \ < \ -3, \ x \ > \ 4\}}\)

6) Solve: \(x^2 \ - \ 1 \ > \ 0\)

\(\color{red}{x_1 \ = \ -1, \ x_2 \ = \ 1}\)
Quadratic_Inequalities6
Test points: \(\color{red}{x \ = \ -2, \ x \ = \ 0, x \ = \ 2}\)
So, \(\color{red}{⇒ \ x \ < \ -1, \ x \ > \ 1}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ x \ < \ -1, \ x \ > \ 1\}}\)

7) Solve: \(x^2 \ - \ 4 \ < \ 0\)

\(\color{red}{x_1 \ = \ -2, \ x_2 \ = \ 2}\)
Quadratic Inequalities7
Test points: \(\color{red}{x \ = \ -3, \ x \ = \ 0, x \ = \ 3}\)
So, \(\color{red}{⇒ \ -2 \ < \ x \ < \ 2}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ -2 \ < \ x \ < \ 2\}}\)

8) Solve: \((2x \ + \ 2)(3x \ - \ 6) \ < \ 0\)

\(\color{red}{x_1 \ = \ -1, \ x_2 \ = \ 2}\)
Quadratic Inequalities
Test points: \(\color{red}{x \ = \ -2, \ x \ = \ 0, x \ = \ 3}\)
So, \(\color{red}{⇒ \ -1 \ < \ x \ < \ 2}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ -1 \ < \ x \ < \ 2\}}\)

9) Solve: \(x^2 \ - \ x \ - \ 12 \ > \ 0\)

\(\color{red}{x_1 \ = \ -3, \ x_2 \ = \ 4}\)
Quadratic Inequalities2
Test points: \(\color{red}{x \ = \ -4, \ x \ = \ 0, x \ = \ 5}\)
So, \(\color{red}{⇒ \ x \ < \ -3, \ x \ > \ 4}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ x \ < \ -3, \ x \ > \ 4\}}\)

10) Solve: \((-x \ - \ 1)(x \ + \ 3) \ > \ 0\)

\(\color{red}{x_1 \ = \ -1, \ x_2 \ = \ -3}\)
Quadratic Inequalities5
Test points: \(\color{red}{x \ = \ -4, \ x \ = \ -2, x \ = \ 0}\)
So, \(\color{red}{⇒ \ -3 \ < \ x \ < \ -1}\) is the solution.
Solution set: \(\color{red}{\{x \ | \ -3 \ < \ x \ < \ -1\}}\)

Solve Quadratic Inequalities Practice Quiz