Solve the Inequality for x -2x^2+8x+17<-7

$- 2 x^{2} + 8 x + 17 < - 7$

Convert the inequality to an equation.

$- 2 x^{2} + 8 x + 17 = - 7$

Add $7$ to both sides of the equation.

$- 2 x^{2} + 8 x + 17 + 7 = 0$

Add $17$ and $7$ .

$- 2 x^{2} + 8 x + 24 = 0$

Factor the left side of the equation.

Factor $- 2$ out of $- 2 x^{2} + 8 x + 24$ .

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

$- 2 x^{2} + 8 x + 24 = 0$

Factor $- 2$ out of $8 x$ .

$- 2 x^{2} - 2 (- 4 x) + 24 = 0$

Factor $- 2$ out of $24$ .

$- 2 x^{2} - 2 (- 4 x) - 2 \cdot - 12 = 0$

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 4 x)$ .

$- 2 (x^{2} - 4 x) - 2 \cdot - 12 = 0$

Factor $- 2$ out of $- 2 (x^{2} - 4 x) - 2 (- 12)$ .

$- 2 (x^{2} - 4 x - 12) = 0$

Factor.

Factor $x^{2} - 4 x - 12$ 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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Write the factored form using these integers.

$- 2 ((x - 6) (x + 2)) = 0$

Remove unnecessary parentheses.

$- 2 (x - 6) (x + 2) = 0$

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

$x - 6 = 0$

$x + 2 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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 $- 2 (x - 6) (x + 2) = 0$ true.

$x = 6, - 2$

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 6$

$x > 6$

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.

$- 2 {(- 4)}^{2} + 8 (- 4) + 17 < - 7$

The left side $- 47$ is less than the right side $- 7$ , which means that the given statement is always true.

True

Test a value on the interval $- 2 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

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

$- 2 {(0)}^{2} + 8 (0) + 17 < - 7$

The left side $17$ is not less than the right side $- 7$ , which means that the given statement is false.

False

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$- 2 {(8)}^{2} + 8 (8) + 17 < - 7$

The left side $- 47$ is less than the right side $- 7$ , which means that the given statement is always true.

True

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

$x < - 2$ True

$- 2 < x < 6$ False

$x > 6$ True

$x < - 2$ True

$- 2 < x < 6$ False

$x > 6$ True

The solution consists of all of the true intervals.

$x < - 2$ or $x > 6$

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 6$

Interval Notation:

$(- \infty, - 2) \cup (6, \infty)$

Solve the Inequality for x -2x^2+8x+17<-7

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