Solve the Inequality for x x^2+16x+24>6x

$x^{2} + 16 x + 24 > 6 x$

Step 1

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

Subtract $6 x$ from both sides of the inequality.

$x^{2} + 16 x + 24 - 6 x > 0$

Subtract $6 x$ from $16 x$ .

$x^{2} + 10 x + 24 > 0$

Step 2

Convert the inequality to an equation.

$x^{2} + 10 x + 24 = 0$

Step 3

Factor $x^{2} + 10 x + 24$ 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 $24$ and whose sum is $10$ .

$4, 6$

Write the factored form using these integers.

$(x + 4) (x + 6) = 0$

Step 4

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

$x + 4 = 0$

$x + 6 = 0$

Step 5

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 6

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 7

The final solution is all the values that make $(x + 4) (x + 6) = 0$ true.

$x = - 4, - 6$

Step 8

Use each root to create test intervals.

$x < - 6$

$- 6 < x < - 4$

$x > - 4$

Step 9

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 $x < - 6$ 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.

${(- 8)}^{2} + 16 (- 8) + 24 > 6 (- 8)$

The left side $- 40$ is greater than the right side $- 48$ , which means that the given statement is always true.

True

Test a value on the interval $- 6 < x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < - 4$ and see if this value makes the original inequality true.

$x = - 5$

Replace $x$ with $- 5$ in the original inequality.

${(- 5)}^{2} + 16 (- 5) + 24 > 6 (- 5)$

The left side $- 31$ is not greater than the right side $- 30$ , 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.

$x = 0$

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

${(0)}^{2} + 16 (0) + 24 > 6 (0)$

The left side $24$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - 6$ True

$- 6 < x < - 4$ False

$x > - 4$ True

$x < - 6$ True

$- 6 < x < - 4$ False

$x > - 4$ True

Step 10

The solution consists of all of the true intervals.

$x < - 6$ or $x > - 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < - 6 or x > - 4$

Interval Notation:

$(- \infty, - 6) \cup (- 4, \infty)$

Step 12

Solve the Inequality for x x^2+16x+24>6x

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