Solve each quadratic inequality. Write each solution set in interval notation. See Example...

Solve each quadratic inequality. Write each solution set in interval notation. See Example 1 and Example 2.

EXAMPLE

Solving a Quadratic Inequality

Solve x2 − x − 12 < 0.

SOLUTION

Step 1 Find the values of x that satisfy x2− x − 12 = 0.

Step 2 The two numbers −3 and 4 cause the expression x2 − x − 12 to equalzero and can be used to divide the number line into three intervals, as shown in Figure 1. The expression x2 − x − 12 will take on a value that is either lessthan zero or greaterthan zero on each of these intervals. Since we are looking for x-values that make the expression lessthan zero, use open circles at −3 and 4 to indicate that they are not included in the solution set.

Figure

Step 3 Choose a test value in each interval to see whether it satisfies the original inequality, x2 − x − 12 < 0. If the test value makes the statement true, then the entire interval belongs to the solution set.

Since the values in Interval B make the inequality true, the solution set is (−3, 4). See Figure 2.

Figure

EXAMPLE

Solving a Quadratic Inequality

Solve 2x2 + 5x −12 ≥ 0.

SOLUTION

Step 1 Find the values of x that satisfy 2x2+ 5x −12 = 0.

Step 2 The values and −4 cause the inequality 2x2+ 5x −12 to equal 0 and can be used to form the intervals (− ∞, −4), 1 , and on the number line, as seen in Figure 1.

Figure

Step 3 Choose a test value from each interval.

The values in Intervals A and C make the inequality true, so the solution set is the union (Section R.1) of the two intervals, written (− ∞, − 4] ∪ .The graph of the solution set is shown in Figure 2.

Figure

x2 + 5x + 7 < 0