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

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

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Solve advanced linear inequalities

Full solution

Q.

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

Identify Zeros: Identify the zeros of the function by setting each factor equal to zero; $x^2 = 0$ , $x - 2 = 0$ , and $(x + 3)^2 = 0$ .
Solve for x: Solve for x in each equation; $x = 0$ , $x = 2$ , and $x = -3$ .
Plot on Number Line: Plot the zeros on a number line and determine the intervals to test; the zeros 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 original inequality; for example, choose $-4$ , $-1$ , $1$ , and $3$ .
Test $x = -4$ : Test $x = -4$ in the inequality; $(-4)^2(-4 - 2)(-4 + 3)^2 < 0$ turns out to be $16(-6)(-1)^2 < 0$ , which simplifies to $96 < 0$ , which is false.
Test $x = -1$ : Test $x = -1$ in the inequality; $(-1)^2(-1 - 2)(-1 + 3)^2 < 0$ turns out to be $1(-3)(2)^2 < 0$ , which simplifies to $-12 < 0$ , which is true.
Test $x = 1$ : Test $x = 1$ in the inequality; $(1)^2(1 - 2)(1 + 3)^2 < 0$ turns out to be $1(-1)(4)^2 < 0$ , which simplifies to $-16 < 0$ , which is true.

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