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$x^2(x-2)(x+3)^2<0$

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Solve advanced linear inequalities

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Q.

x^2(x-2)(x+3)^2<0

Identify Critical Points: Identify the critical points of the inequality by setting each factor equal to zero. $\newline$ $x^2 = 0$ gives $x = 0$ $\newline$ $(x - 2) = 0$ gives $x = 2$ $\newline$ $(x + 3)^2 = 0$ gives $x = -3$
Plot on Number Line: Plot the critical points on a number line and determine the intervals to test. $\newline$ The critical points divide the number line into four intervals: $(-\infty, -3)$ , $(-3, 0)$ , $(0, 2)$ , and $(2, \infty)$ .
Choose Test Points: Choose test points from each interval and plug them into the inequality to determine the sign of the expression in that interval. $\newline$ For $(-\infty, -3)$ , choose $x = -4$ : $(-4)^2(-4 - 2)(-4 + 3)^2 > 0$ , which is positive. $\newline$ For $(-3, 0)$ , choose $x = -1$ : $(-1)^2(-1 - 2)(-1 + 3)^2 < 0$ , which is negative. $\newline$ For $(0, 2)$ , choose $x = 1$ : $(1)^2(1 - 2)(1 + 3)^2 < 0$ , which is negative. $\newline$ For $(2, \infty)$ , choose $x = -4$ $0$ : $x = -4$ $1$ , which is positive.
Determine Satisfied Intervals: Determine the intervals where the inequality is satisfied. $\newline$ The inequality is satisfied in the intervals where the expression is negative, which are $(-3, 0)$ and $(0, 2)$ .
Consider Critical Points: Consider the behavior at the critical points. Since the inequality is strict ( $<0$ ), the critical points themselves are not included in the solution.
Combine Intervals: Combine the intervals to express the solution. $\newline$ The solution to the inequality is the union of the intervals where the expression is negative, which is $(-3, 0) \cup (0, 2)$ .

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