Solve the Inequality for x (x-1)^2(x+2)(x-3)<0

${(x - 1)}^{2} (x + 2) (x - 3) < 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 1)}^{2} = 0$

$x + 2 = 0$

$x - 3 = 0$

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{2}$ equal to $0$ .

${(x - 1)}^{2} = 0$

Solve ${(x - 1)}^{2} = 0$ for $x$ .

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make ${(x - 1)}^{2} (x + 2) (x - 3) < 0$ true.

$x = 1, - 2, 3$

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 1$

$1 < x < 3$

$x > 3$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

${((- 4) - 1)}^{2} ((- 4) + 2) ((- 4) - 3) < 0$

The left side $350$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

${((0) - 1)}^{2} ((0) + 2) ((0) - 3) < 0$

The left side $- 6$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

${((2) - 1)}^{2} ((2) + 2) ((2) - 3) < 0$

The left side $- 4$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

${((6) - 1)}^{2} ((6) + 2) ((6) - 3) < 0$

The left side $600$ is not less than the right side $0$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 2$ False

$- 2 < x < 1$ True

$1 < x < 3$ True

$x > 3$ False

$x < - 2$ False

$- 2 < x < 1$ True

$1 < x < 3$ True

$x > 3$ False

The solution consists of all of the true intervals.

$- 2 < x < 1$ or $1 < x < 3$

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 1 or 1 < x < 3$

Interval Notation:

$(- 2, 1) \cup (1, 3)$

Solve the Inequality for x (x-1)^2(x+2)(x-3)<0

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