Solve the Inequality for x 2x-x^2<-8

$2 x - x^{2} < - 8$

Convert the inequality to an equation.

$2 x - x^{2} = - 8$

Add $8$ to both sides of the equation.

$2 x - x^{2} + 8 = 0$

Factor the left side of the equation.

Factor $- 1$ out of $2 x - x^{2} + 8$ .

Reorder $2 x$ and $- x^{2}$ .

$- x^{2} + 2 x + 8 = 0$

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

$- (x^{2}) + 2 x + 8 = 0$

Factor $- 1$ out of $2 x$ .

$- (x^{2}) - (- 2 x) + 8 = 0$

Rewrite $8$ as $- 1 (- 8)$ .

$- (x^{2}) - (- 2 x) - 1 \cdot - 8 = 0$

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

$- (x^{2} - 2 x) - 1 \cdot - 8 = 0$

Factor $- 1$ out of $- (x^{2} - 2 x) - 1 (- 8)$ .

$- (x^{2} - 2 x - 8) = 0$

Factor.

Factor $x^{2} - 2 x - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

$- ((x - 4) (x + 2)) = 0$

Remove unnecessary parentheses.

$- (x - 4) (x + 2) = 0$

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

$x - 4 = 0$

$x + 2 = 0$

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.

$x = 4$

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.

$x = - 2$

The final solution is all the values that make $- (x - 4) (x + 2) = 0$ true.

$x = 4, - 2$

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 4$

$x > 4$

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.

$2 (- 4) - {(- 4)}^{2} < - 8$

The left side $- 24$ is less than the right side $- 8$ , which means that the given statement is always true.

True

Test a value on the interval $- 2 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$2 (0) - {(0)}^{2} < - 8$

The left side $0$ is not less than the right side $- 8$ , which means that the given statement is false.

False

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.

$2 (6) - {(6)}^{2} < - 8$

The left side $- 24$ is less than the right side $- 8$ , which means that the given statement is always true.

True

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

$x < - 2$ True

$- 2 < x < 4$ False

$x > 4$ True

$x < - 2$ True

$- 2 < x < 4$ False

$x > 4$ True

The solution consists of all of the true intervals.

$x < - 2$ or $x > 4$

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 4$

Interval Notation:

$(- \infty, - 2) \cup (4, \infty)$

Solve the Inequality for x 2x-x^2<-8

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