Solve the Inequality for x x^2-4x-45>=0

$x^{2} - 4 x - 45 \geq 0$

Convert the inequality to an equation.

$x^{2} - 4 x - 45 = 0$

Factor $x^{2} - 4 x - 45$ 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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Write the factored form using these integers.

$(x - 9) (x + 5) = 0$

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

$x - 9 = 0$

$x + 5 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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 $(x - 9) (x + 5) = 0$ true.

$x = 9, - 5$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 9$

$x > 9$

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 $x < - 5$ 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.

${(- 8)}^{2} - 4 \cdot - 8 - 45 \geq 0$

The left side $51$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 5 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < 9$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} - 4 \cdot 0 - 45 \geq 0$

The left side $- 45$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

${(12)}^{2} - 4 \cdot 12 - 45 \geq 0$

The left side $51$ 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.

$x < - 5$ True

$- 5 < x < 9$ False

$x > 9$ True

$x < - 5$ True

$- 5 < x < 9$ False

$x > 9$ True

The solution consists of all of the true intervals.

$x \leq - 5$ or $x \geq 9$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 5 or x \geq 9$

Interval Notation:

$(- \infty, - 5] \cup [9, \infty)$

Solve the Inequality for x x^2-4x-45>=0

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