POLYNOMIAL FUNCTIONS Solving Polynomial Inequalities (Page 2 of 2)

We now interpret the inequality 9x^5-12x^4-5x^3+66x^2-76x+24 > 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.

 TRACE key. Left arrow key....

 Right arrow key....

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

Since the x-intercepts have a y-value equal to 0, they are NOT part of the solution set. The x-values to the left of -2 yield negative y-values. The values of x between -2 and 2/3, and to the right of 2/3, produce positive y-values. The inequality 9x^5-12x^4-5x^3+66x^2-76x+24 > 0 means that the graph of Y1 must lie above the x-axis. Looking at the graph, we see this is true if x is to the right of -2 but not including x = 2/3. So, the solution set for the inequality is {x: x > -2 and x is not equal to 2/3}, or in interval notation, (-2, 2/3)U(2/3, infinity).
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