Solve the Inequality for x (x-5)(x+3)<=0

$(x - 5) (x + 3) \leq 0$

Step 1

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

$x - 5 = 0$

$x + 3 = 0$

Step 2

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 3

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4

The final solution is all the values that make $(x - 5) (x + 3) \leq 0$ true.

$x = 5, - 3$

Step 5

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 5$

$x > 5$

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 $x < - 3$ 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.

$((- 6) - 5) ((- 6) + 3) \leq 0$

The left side $33$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 3 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < 5$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) - 5) ((0) + 3) \leq 0$

The left side $- 15$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$((8) - 5) ((8) + 3) \leq 0$

The left side $33$ is greater 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.

$x < - 3$ False

$- 3 < x < 5$ True

$x > 5$ False

$x < - 3$ False

$- 3 < x < 5$ True

$x > 5$ False

Step 7

The solution consists of all of the true intervals.

$- 3 \leq x \leq 5$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x \leq 5$

Interval Notation:

$[- 3, 5]$

Step 9

Solve the Inequality for x (x-5)(x+3)<=0

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