Solve the Inequality for x -x^2-2x<5x-8

$- x^{2} - 2 x < 5 x - 8$

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

Subtract $5 x$ from both sides of the inequality.

$- x^{2} - 2 x - 5 x < - 8$

Subtract $5 x$ from $- 2 x$ .

$- x^{2} - 7 x < - 8$

Convert the inequality to an equation.

$- x^{2} - 7 x = - 8$

Add $8$ to both sides of the equation.

$- x^{2} - 7 x + 8 = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - 7 x + 8$ .

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

$- (x^{2}) - 7 x + 8 = 0$

Factor $- 1$ out of $- 7 x$ .

$- (x^{2}) - (7 x) + 8 = 0$

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

$- (x^{2}) - (7 x) - 1 \cdot - 8 = 0$

Factor $- 1$ out of $- (x^{2}) - (7 x)$ .

$- (x^{2} + 7 x) - 1 \cdot - 8 = 0$

Factor $- 1$ out of $- (x^{2} + 7 x) - 1 (- 8)$ .

$- (x^{2} + 7 x - 8) = 0$

Factor.

Factor $x^{2} + 7 x - 8$ 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 $7$ .

$- 1, 8$

Write the factored form using these integers.

$- ((x - 1) (x + 8)) = 0$

Remove unnecessary parentheses.

$- (x - 1) (x + 8) = 0$

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

$x - 1 = 0$

$x + 8 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $- (x - 1) (x + 8) = 0$ true.

$x = 1, - 8$

Use each root to create test intervals.

$x < - 8$

$- 8 < x < 1$

$x > 1$

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 $x < - 8$ to see if it makes the inequality true.

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

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

$- {(- 10)}^{2} - 2 \cdot - 10 < 5 (- 10) - 8$

The left side $- 80$ is less than the right side $- 58$ , which means that the given statement is always true.

True

Test a value on the interval $- 8 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

$- {(0)}^{2} - 2 \cdot 0 < 5 (0) - 8$

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 $x > 1$ to see if it makes the inequality true.

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$- {(4)}^{2} - 2 \cdot 4 < 5 (4) - 8$

The left side $- 24$ is less than the right side $12$ , which means that the given statement is always true.

True

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

$x < - 8$ True

$- 8 < x < 1$ False

$x > 1$ True

$x < - 8$ True

$- 8 < x < 1$ False

$x > 1$ True

The solution consists of all of the true intervals.

$x < - 8$ or $x > 1$

The result can be shown in multiple forms.

Inequality Form:

$x < - 8 or x > 1$

Interval Notation:

$(- \infty, - 8) \cup (1, \infty)$

Solve the Inequality for x -x^2-2x<5x-8

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