Solve the Inequality for x x^2>4(x+8)

$x^{2} > 4 (x + 8)$

Step 1

Simplify $4 (x + 8)$ .

Apply the distributive property.

$x^{2} > 4 x + 4 \cdot 8$

Multiply $4$ by $8$ .

$x^{2} > 4 x + 32$

Step 2

Subtract $4 x$ from both sides of the inequality.

$x^{2} - 4 x > 32$

Step 3

Convert the inequality to an equation.

$x^{2} - 4 x = 32$

Step 4

Subtract $32$ from both sides of the equation.

$x^{2} - 4 x - 32 = 0$

Step 5

Factor $x^{2} - 4 x - 32$ 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 $- 32$ and whose sum is $- 4$ .

$- 8, 4$

Write the factored form using these integers.

$(x - 8) (x + 4) = 0$

Step 6

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

$x - 8 = 0$

$x + 4 = 0$

Step 7

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

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

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Step 8

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 9

The final solution is all the values that make $(x - 8) (x + 4) = 0$ true.

$x = 8, - 4$

Step 10

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 8$

$x > 8$

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 $x < - 4$ 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.

${(- 6)}^{2} > 4 ((- 6) + 8)$

The left side $36$ is greater than the right side $8$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < 8$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 8$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} > 4 ((0) + 8)$

The left side $0$ is not greater than the right side $32$ , which means that the given statement is false.

False

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.

${(10)}^{2} > 4 ((10) + 8)$

The left side $100$ is greater than the right side $72$ , which means that the given statement is always true.

True

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

$x < - 4$ True

$- 4 < x < 8$ False

$x > 8$ True

$x < - 4$ True

$- 4 < x < 8$ False

$x > 8$ True

Step 12

The solution consists of all of the true intervals.

$x < - 4$ or $x > 8$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x < - 4 or x > 8$

Interval Notation:

$(- \infty, - 4) \cup (8, \infty)$

Step 14

Solve the Inequality for x x^2>4(x+8)

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