Quadratic Inequalities That Cannot Be Factored
There is another method, called the test point method, that can be used instead of the
sign graph to solve inequalities.
The following example shows how to solve a quadratic inequality that involves a
prime polynomial.
Example 1
Solving a quadratic inequality using the quadratic formula
Solve x^{2}  4x  6 > 0 and graph the solution set.
Solution
The quadratic polynomial is prime, but we can solve x^{2}  4x  6
= 0 by the quadratic
formula:
As in the previous examples, the solutions to the equation divide the number line
into the intervals (∞, 2  ), (2
,
2 +), and (2
+,
∞) on
which the quadratic polynomial has either a positive or negative value. To determine
which, we select an arbitrary test point in each interval. Because 2 +
≈
5.2 and 2 
≈ 1.2, we choose a test point that is less than
1.2, one between 1.2 and 5.2, and one that is greater than 5.2. We have selected
2, 0, and 7 for test points, as shown in the figure below.
Now evaluate x^{2}  4x  6 at
each test point.
Test point 
Value of x^{2}  4x  6 at the test point 
Sign of x^{2}  4x  6 in interval of test point 
2 
6 
Positive 
0 
6 
Negative 
7 
15 
Positive 
Because x^{2}  4x  6 is positive at the test points 2 and 7, it is positive at every
point in the intervals containing those test points. So the solution set to the inequality x^{2}
 4x  6 > 0 is
and its graph is shown in the figure below.
The test point method used in Example 1 can be used also on inequalities that do
factor. We summarize the strategy for solving inequalities using test points
below.
Strategy for Solving Quadratic Inequalities
1. Rewrite the inequality with 0 on the right.
2. Solve the quadratic equation that results from replacing the inequality
symbol with the equals symbol.
3. Locate the solutions to the quadratic equation on a number line.
4. Select a test point in each interval determined by the solutions to the
quadratic equation.
5. Test each point in the original quadratic inequality to determine which
intervals satisfy the inequality.
