Solve the Inequality for x -x^2-10x-21>=0

$- x^{2} - 10 x - 21 \geq 0$

Step 1

Convert the inequality to an equation.

$- x^{2} - 10 x - 21 = 0$

Step 2

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - 10 x - 21$ .

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

$- (x^{2}) - 10 x - 21 = 0$

Factor $- 1$ out of $- 10 x$ .

$- (x^{2}) - (10 x) - 21 = 0$

Rewrite $- 21$ as $- 1 (21)$ .

$- (x^{2}) - (10 x) - 1 \cdot 21 = 0$

Factor $- 1$ out of $- (x^{2}) - (10 x)$ .

$- (x^{2} + 10 x) - 1 \cdot 21 = 0$

Factor $- 1$ out of $- (x^{2} + 10 x) - 1 (21)$ .

$- (x^{2} + 10 x + 21) = 0$

Factor.

Factor $x^{2} + 10 x + 21$ 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 $21$ and whose sum is $10$ .

$3, 7$

Write the factored form using these integers.

$- ((x + 3) (x + 7)) = 0$

Remove unnecessary parentheses.

$- (x + 3) (x + 7) = 0$

Step 3

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

$x + 3 = 0$

$x + 7 = 0$

Step 4

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 5

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 6

The final solution is all the values that make $- (x + 3) (x + 7) = 0$ true.

$x = - 3, - 7$

Step 7

Use each root to create test intervals.

$x < - 7$

$- 7 < x < - 3$

$x > - 3$

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 $x < - 7$ to see if it makes the inequality true.

Choose a value on the interval $x < - 7$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

$- {(- 10)}^{2} - 10 \cdot - 10 - 21 \geq 0$

The left side $- 21$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 7 < x < - 3$ to see if it makes the inequality true.

Choose a value on the interval $- 7 < x < - 3$ and see if this value makes the original inequality true.

$x = - 5$

Replace $x$ with $- 5$ in the original inequality.

$- {(- 5)}^{2} - 10 \cdot - 5 - 21 \geq 0$

The left side $4$ is greater 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.

$x = 0$

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

$- {(0)}^{2} - 10 \cdot 0 - 21 \geq 0$

The left side $- 21$ is 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.

$x < - 7$ False

$- 7 < x < - 3$ True

$x > - 3$ False

$x < - 7$ False

$- 7 < x < - 3$ True

$x > - 3$ False

Step 9

The solution consists of all of the true intervals.

$- 7 \leq x \leq - 3$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 7 \leq x \leq - 3$

Interval Notation:

$[- 7, - 3]$

Step 11

Solve the Inequality for x -x^2-10x-21>=0

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