Solve the Inequality for x x^2+8x+15<0

$x^{2} + 8 x + 15 < 0$

Convert the inequality to an equation.

$x^{2} + 8 x + 15 = 0$

Factor $x^{2} + 8 x + 15$ 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 $15$ and whose sum is $8$ .

$3, 5$

Write the factored form using these integers.

$(x + 3) (x + 5) = 0$

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

$x + 3 = 0$

$x + 5 = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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 $(x + 3) (x + 5) = 0$ true.

$x = - 3, - 5$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 3$

$x > - 3$

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.

${(- 8)}^{2} + 8 (- 8) + 15 < 0$

The left side $15$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < - 3$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < - 3$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

${(- 4)}^{2} + 8 (- 4) + 15 < 0$

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > - 3$ to see if it makes the inequality true.

Choose a value on the interval $x > - 3$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} + 8 (0) + 15 < 0$

The left side $15$ is not less than the right side $0$ , which means that the given statement is false.

False

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

$x < - 5$ False

$- 5 < x < - 3$ True

$x > - 3$ False

$x < - 5$ False

$- 5 < x < - 3$ True

$x > - 3$ False

The solution consists of all of the true intervals.

$- 5 < x < - 3$

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < - 3$

Interval Notation:

$(- 5, - 3)$

Solve the Inequality for x x^2+8x+15<0

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