Solve and Graph Polynomial Inequalities

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Pre-Calculus › Solve and Graph Polynomial Inequalities

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What is the solution to the following inequality?

$x < -\frac{2}{3} \bigcup 1 < x < 2$

$-\frac{2}{3} < x < 1 \bigcup x > 2$

$x < 1 \bigcup x > 2$

$-\frac{2}{3} < x < 1 \bigcup x > 2$

First, we must solve for the roots of the cubic polynomial equation.

We obtain that the roots are $-\frac{2}{3}, 1, 2$ .

Now there are four regions created by these numbers:

$x < -\frac{2}{3}$ . In this region, the values of the polynomial are negative (i.e.plug in and you obtain
$-\frac{2}{3} < x < 1$ . In this region, the values of the polynomial are positive (when , polynomial evaluates to )
. In this region the polynomial switches again to negative.
. In this region the values of the polynomial are positive

Hence the two regions we want are $x < -\frac{2}{3}$ and .