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Solving Quadratic Inequalities

To solve a quadratic inequality, follow these steps:

  1. Solve the inequality as though it were an equation.

    The real solutions to the equation become boundary points for the solution to the inequality.

  2. Make the boundary points solid circles if the original inequality includes equality; otherwise, make the boundary points open circles.

  3. Select points from each of the regions created by the boundary points. Replace these “test points” in the original inequality.

  4. If a test point satisfies the original inequality, then the region that contains that test point is part of the solutions.

  5. Represent the solution in graphic form and in solution set form.

Example 1: Solve

Solve . By the zero product property,




Make the boundary points. Here, the boundary points are open circles because the original inequality does not include equality (see Figure 1 ).





Figure 1

Boundary points.

Select points from the different regions created (see Figure 2 ).





Figure 2

Three regions are created.

Try

See if the test points satisfy the original inequality.




Since satisfies the original inequality, the region is part of the solution. Since does not satisfy the original inequality, the region is not part of the solution. Since satisfies the original inequality, the region is part of the solution.

Represent the solution in graphic form and in solution set form. The graphic form is shown in Figure 3 .





Figure 3

Solution to Example 14.

The solution set form is

Example 2: Solve .

Solve




By factoring,




Mark the boundary points using solid circles, as shown in Figure 4 , since the original inequality includes equality.





Figure 4

Solid dots mean inclusion.

Select points from the regions created (see Figure 5 ).





Figure 5

Regions to test for Example 15.

Try

See if the test points satisfy the original inequality.




Since does not satisfy the original inequality, the region is not part of the solution. Since does satisfy the original inequality, the region is part of the solution. Since does not satisfy the original inequality, the region is not part of the solution.

Represent the solution in graphic form and in solution set form. The graphic form is shown in Figure 6 .





Figure 6

Solution to Example 15.

The set form is

Example 3: Solve .

Solve




Since this quadratic is not easily factorable, the quadratic formula is used to solve it.




Reduce by dividing out the common factor of 4.




Since is approximately 3.2,




Mark the boundary points using open circles, as shown in Figure 7 , since the original inequality does not include equality.





Figure 7

Open dots mean exclusion.

Select points from the different regions created (see Figure 8 ).





Figure 8

Regions to test for Example 16.

Try

See if the test points satisfy the original inequality.




Since does not satisfy the original inequality, the region is not part of the solution. Since does satisfy the original inequality, the region is part of the solution. Since does not satisfy the original inequality, the region is not part of the solution.

Represent the solution in graphic form and in solution set form. The graphic form is shown in Figure 9 .





Figure 9

Solution to Example 16.

The solution set form is

Example 4: Solve .




Since this quadratic is not factorable using rational numbers, the quadratic formula will be used to solve it.




These are imaginary answers and cannot be graphed on a real number line. Therefore, the inequality has no real solutions.

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