Solve for x x^2+10x-24<0

$x^{2} + 10 x - 24 < 0$

Convert the inequality to an equation.

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

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 $10$ .

$- 2, 12$

Write the factored form using these integers.

$(x - 2) (x + 12) = 0$

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

$x - 2 = 0$

$x + 12 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x + 12$ equal to $0$ and solve for $x$ .

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Subtract $12$ from both sides of the equation.

$x = - 12$

The final solution is all the values that make $(x - 2) (x + 12) = 0$ true.

$x = 2, - 12$

Use each root to create test intervals.

$x < - 12$

$- 12 < x < 2$

$x > 2$

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 $x < - 12$ to see if it makes the inequality true.

Choose a value on the interval $x < - 12$ and see if this value makes the original inequality true.

$x = - 14$

Replace $x$ with $- 14$ in the original inequality.

${(- 14)}^{2} + 10 (- 14) - 24 < 0$

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

False

Test a value on the interval $- 12 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 12 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} + 10 (0) - 24 < 0$

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

True

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

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

${(4)}^{2} + 10 (4) - 24 < 0$

The left side $32$ 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.

$x < - 12$ False

$- 12 < x < 2$ True

$x > 2$ False

$x < - 12$ False

$- 12 < x < 2$ True

$x > 2$ False

The solution consists of all of the true intervals.

$- 12 < x < 2$

The result can be shown in multiple forms.

Inequality Form:

$- 12 < x < 2$

Interval Notation:

$(- 12, 2)$

Solve for x x^2+10x-24<0

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