Solve the Inequality for x 2x^2+4x-87<-8x-7

$2 x^{2} + 4 x - 87 < - 8 x - 7$

Step 1

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

Add $8 x$ to both sides of the inequality.

$2 x^{2} + 4 x - 87 + 8 x < - 7$

Add $4 x$ and $8 x$ .

$2 x^{2} + 12 x - 87 < - 7$

Step 2

Convert the inequality to an equation.

$2 x^{2} + 12 x - 87 = - 7$

Step 3

Add $7$ to both sides of the equation.

$2 x^{2} + 12 x - 87 + 7 = 0$

Step 4

Add $- 87$ and $7$ .

$2 x^{2} + 12 x - 80 = 0$

Step 5

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} + 12 x - 80$ .

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

$2 (x^{2}) + 12 x - 80 = 0$

Factor $2$ out of $12 x$ .

$2 (x^{2}) + 2 (6 x) - 80 = 0$

Factor $2$ out of $- 80$ .

$2 x^{2} + 2 (6 x) + 2 \cdot - 40 = 0$

Factor $2$ out of $2 x^{2} + 2 (6 x)$ .

$2 (x^{2} + 6 x) + 2 \cdot - 40 = 0$

Factor $2$ out of $2 (x^{2} + 6 x) + 2 \cdot - 40$ .

$2 (x^{2} + 6 x - 40) = 0$

Factor.

Factor $x^{2} + 6 x - 40$ 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 $- 40$ and whose sum is $6$ .

$- 4, 10$

Write the factored form using these integers.

$2 ((x - 4) (x + 10)) = 0$

Remove unnecessary parentheses.

$2 (x - 4) (x + 10) = 0$

Step 6

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

$x - 4 = 0$

$x + 10 = 0$

Step 7

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 8

Set $x + 10$ equal to $0$ and solve for $x$ .

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 9

The final solution is all the values that make $2 (x - 4) (x + 10) = 0$ true.

$x = 4, - 10$

Step 10

Use each root to create test intervals.

$x < - 10$

$- 10 < x < 4$

$x > 4$

Step 11

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 $x < - 10$ to see if it makes the inequality true.

Choose a value on the interval $x < - 10$ and see if this value makes the original inequality true.

$x = - 12$

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

$2 {(- 12)}^{2} + 4 (- 12) - 87 < - 8 \cdot - 12 - 7$

The left side $153$ is not less than the right side $89$ , which means that the given statement is false.

False

Test a value on the interval $- 10 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 10 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$2 {(0)}^{2} + 4 (0) - 87 < - 8 \cdot 0 - 7$

The left side $- 87$ is less than the right side $- 7$ , 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.

$2 {(6)}^{2} + 4 (6) - 87 < - 8 \cdot 6 - 7$

The left side $9$ is not less than the right side $- 55$ , which means that the given statement is false.

False

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

$x < - 10$ False

$- 10 < x < 4$ True

$x > 4$ False

$x < - 10$ False

$- 10 < x < 4$ True

$x > 4$ False

Step 12

The solution consists of all of the true intervals.

$- 10 < x < 4$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 10 < x < 4$

Interval Notation:

$(- 10, 4)$

Step 14

Solve the Inequality for x 2x^2+4x-87<-8x-7

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