EXPONENTIAL FUNCTIONS Solving Exponential Inequalities (Page 2 of 2)

We now interpret the inequality 3^(2x+1)-5^(x+2) > 0 in terms of the function that is graphed. The solution set will be all values of x for which the function Y1 has positive y-values. Let's use TRACE to identify where these values of x are (from our previous work, this function and the appropriate WINDOW settings should still be stored in the calculator).

 TRACE key. Left arrow key....

 Right arrow key....

 Calculator screen image. Calculator screen image. Calculator screen image.

Since the x-intercept has a y-value equal to 0, it is NOT part of the solution set. The x-values to the left of 3.607 yield negative y-values. For x-values to the right of 3.607 we obtain positive y-values. The inequality
 3^(2x+1)-5^(x+2) > 0 means that the graph of Y1 must lie above the x-axis. Observing the graph, we see this is true if x is to the right of 3.607. So, the approximate solution set is {x: x>3.607}, or in interval notation, (3.607, infinity).
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