Graph x^2+10x>=-24

$x^{2} + 10 x \geq - 24$

Convert the inequality to an equation.

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

Add $24$ to both sides of the 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 $10$ .

$4, 6$

Write the factored form using these integers.

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

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

$x + 4 = 0$

$x + 6 = 0$

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

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

$x = - 4, - 6$

Use each root to create test intervals.

$x < - 6$

$- 6 < x < - 4$

$x > - 4$

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 $x < - 6$ 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.

${(- 8)}^{2} + 10 (- 8) \geq - 24$

The left side $- 16$ is greater than the right side $- 24$ , 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.

${(- 5)}^{2} + 10 (- 5) \geq - 24$

The left side $- 25$ is less than the right side $- 24$ , 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.

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

The left side $0$ is greater than the right side $- 24$ , 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

The solution consists of all of the true intervals.

$x \leq - 6$ or $x \geq - 4$

Graph x^2+10x>=-24

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