Solve the Inequality for x x^2+9x+13>-7

$x^{2} + 9 x + 13 > - 7$

Convert the inequality to an equation.

$x^{2} + 9 x + 13 = - 7$

Add $7$ to both sides of the equation.

$x^{2} + 9 x + 13 + 7 = 0$

Add $13$ and $7$ .

$x^{2} + 9 x + 20 = 0$

Factor $x^{2} + 9 x + 20$ 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 $20$ and whose sum is $9$ .

$4, 5$

Write the factored form using these integers.

$(x + 4) (x + 5) = 0$

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

$x + 4 = 0$

$x + 5 = 0$

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.

$x = - 4$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $(x + 4) (x + 5) = 0$ true.

$x = - 4, - 5$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 4$

$x > - 4$

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 $x < - 5$ to see if it makes the inequality true.

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

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

${(- 8)}^{2} + 9 (- 8) + 13 > - 7$

The left side $5$ is greater than the right side $- 7$ , which means that the given statement is always true.

True

Test a value on the interval $- 5 < x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < - 4$ and see if this value makes the original inequality true.

$x = - 4.5$

Replace $x$ with $- 4.5$ in the original inequality.

${(- 4.5)}^{2} + 9 (- 4.5) + 13 > - 7$

The left side $- 7.25$ is not greater than the right side $- 7$ , 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.

${(0)}^{2} + 9 (0) + 13 > - 7$

The left side $13$ is greater 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.

$x < - 5$ True

$- 5 < x < - 4$ False

$x > - 4$ True

$x < - 5$ True

$- 5 < x < - 4$ False

$x > - 4$ True

The solution consists of all of the true intervals.

$x < - 5$ or $x > - 4$

The result can be shown in multiple forms.

Inequality Form:

$x < - 5 or x > - 4$

Interval Notation:

$(- \infty, - 5) \cup (- 4, \infty)$

Solve the Inequality for x x^2+9x+13>-7

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