Solve the Inequality for x -x^2-9x+36>=-4x

$- x^{2} - 9 x + 36 \geq - 4 x$

Step 1

Move all terms containing $x$ to the left side of the inequality.

Add $4 x$ to both sides of the inequality.

$- x^{2} - 9 x + 36 + 4 x \geq 0$

Add $- 9 x$ and $4 x$ .

$- x^{2} - 5 x + 36 \geq 0$

Step 2

Convert the inequality to an equation.

$- x^{2} - 5 x + 36 = 0$

Step 3

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - 5 x + 36$ .

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 5 x + 36 = 0$

Factor $- 1$ out of $- 5 x$ .

$- (x^{2}) - (5 x) + 36 = 0$

Rewrite $36$ as $- 1 (- 36)$ .

$- (x^{2}) - (5 x) - 1 \cdot - 36 = 0$

Factor $- 1$ out of $- (x^{2}) - (5 x)$ .

$- (x^{2} + 5 x) - 1 \cdot - 36 = 0$

Factor $- 1$ out of $- (x^{2} + 5 x) - 1 (- 36)$ .

$- (x^{2} + 5 x - 36) = 0$

Factor.

Factor $x^{2} + 5 x - 36$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 36$ and whose sum is $5$ .

$- 4, 9$

Write the factored form using these integers.

$- ((x - 4) (x + 9)) = 0$

Remove unnecessary parentheses.

$- (x - 4) (x + 9) = 0$

Step 4

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

$x - 4 = 0$

$x + 9 = 0$

Step 5

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 6

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 7

The final solution is all the values that make $- (x - 4) (x + 9) = 0$ true.

$x = 4, - 9$

Step 8

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 4$

$x > 4$

Step 9

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 $x < - 9$ to see if it makes the inequality true.

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Replace $x$ with $- 12$ in the original inequality.

$- {(- 12)}^{2} - 9 \cdot - 12 + 36 \geq - 4 \cdot - 12$

The left side $0$ is less than the right side $48$ , which means that the given statement is false.

False

Test a value on the interval $- 9 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 9 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$- {(0)}^{2} - 9 \cdot 0 + 36 \geq - 4 \cdot 0$

The left side $36$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

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

$- {(6)}^{2} - 9 \cdot 6 + 36 \geq - 4 \cdot 6$

The left side $- 54$ is less than the right side $- 24$ , which means that the given statement is false.

False

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

$x < - 9$ False

$- 9 < x < 4$ True

$x > 4$ False

$x < - 9$ False

$- 9 < x < 4$ True

$x > 4$ False

Step 10

The solution consists of all of the true intervals.

$- 9 \leq x \leq 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- 9 \leq x \leq 4$

Interval Notation:

$[- 9, 4]$

Step 12

Solve the Inequality for x -x^2-9x+36>=-4x

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