Solve the Inequality for x (x-3)(x+2)>0

$(x - 3) (x + 2) > 0$

Step 1

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

$x - 3 = 0$

$x + 2 = 0$

Step 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.

$x = 3$

Step 3

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.

$x = - 2$

Step 4

The final solution is all the values that make $(x - 3) (x + 2) > 0$ true.

$x = 3, - 2$

Step 5

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 3$

$x > 3$

Step 6

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.

$((- 4) - 3) ((- 4) + 2) > 0$

The left side $14$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 2 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) - 3) ((0) + 2) > 0$

The left side $- 6$ is not greater than the right side $0$ , which means that the given statement is false.

False

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.

$((6) - 3) ((6) + 2) > 0$

The left side $24$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - 2$ True

$- 2 < x < 3$ False

$x > 3$ True

$x < - 2$ True

$- 2 < x < 3$ False

$x > 3$ True

Step 7

The solution consists of all of the true intervals.

$x < - 2$ or $x > 3$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 3$

Interval Notation:

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

Step 9

Solve the Inequality for x (x-3)(x+2)>0

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