Solve the Inequality for x x^2+5x<=0

$x^{2} + 5 x \leq 0$

Step 1

Convert the inequality to an equation.

$x^{2} + 5 x = 0$

Step 2

Factor $x$ out of $x^{2} + 5 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x + 5 x = 0$

Factor $x$ out of $5 x$ .

$x \cdot x + x \cdot 5 = 0$

Factor $x$ out of $x \cdot x + x \cdot 5$ .

$x (x + 5) = 0$

Step 3

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

$x = 0$

$x + 5 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6

The final solution is all the values that make $x (x + 5) = 0$ true.

$x = 0, - 5$

Step 7

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 0$

$x > 0$

Step 8

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 $x < - 5$ 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.

${(- 8)}^{2} + 5 (- 8) \leq 0$

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

False

Test a value on the interval $- 5 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

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

${(- 2)}^{2} + 5 (- 2) \leq 0$

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 $x > 0$ to see if it makes the inequality true.

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

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

${(2)}^{2} + 5 (2) \leq 0$

The left side $14$ 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.

$x < - 5$ False

$- 5 < x < 0$ True

$x > 0$ False

$x < - 5$ False

$- 5 < x < 0$ True

$x > 0$ False

Step 9

The solution consists of all of the true intervals.

$- 5 \leq x \leq 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x \leq 0$

Interval Notation:

$[- 5, 0]$

Step 11

Solve the Inequality for x x^2+5x<=0

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