In Example 2, change the − sign before the 3 to + and change 2x to 4x, then solve the resulting inequality, and graph the solution.

Data from Example 2

Solve the inequality x² − 3 > 2x.
We first find the equivalent inequality with zero on the right. Therefore, we have x² − 2x − 3 > 0. We then factor the left member and have (x + 1)(x − 3) > 0. Setting each factor equal to zero, we find the critical values are −1 and 3. These critical values are the only possible places where the function can change in sign. Therefore, we must determine the sign of f(x) for each of the intervals x < −1, −1< x < 3, and x > 3.

To do this, we will choose a test value from each interval and substitute it into the function to determine its sign. The following table shows each interval, the test value, the sign of each factor on each interval, and the resulting sign of the function.

Because we want values for which the product is greater than zero (or positive), the solution is x < −1 or x > 3. The solution that is shown in Fig. 17.20(b) corresponds to the positive values of the function f(x) = x² − 2x − 3, shown in Fig. 17.20(a).

Transcribed Image Text:

Interval x < -1 -1 < x < 3 x > 3 Test value -2 0 4 (x + 1)(x − 3) + + + Sign of (x + 1)(x − 3) + +