Solve the Inequality for x (x^2+3x-18)/(x-7)>=0

$\frac{x^{2} + 3 x - 18}{x - 7} \geq 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} + 3 x - 18 = 0$

$x - 7 = 0$

Step 2

Factor $x^{2} + 3 x - 18$ 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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Write the factored form using these integers.

$(x - 3) (x + 6) = 0$

Step 3

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

$x - 3 = 0$

$x + 6 = 0$

Step 4

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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.

$x = - 6$

Step 6

The final solution is all the values that make $(x - 3) (x + 6) = 0$ true.

$x = 3, - 6$

Step 7

Add $7$ to both sides of the equation.

$x = 7$

Step 8

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 3, - 6$

$x = 7$

Step 9

Consolidate the solutions.

$x = 3, - 6, 7$

Step 10

Find the domain of $\frac{x^{2} + 3 x - 18}{x - 7}$ .

Set the denominator in $\frac{x^{2} + 3 x - 18}{x - 7}$ equal to $0$ to find where the expression is undefined.

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The domain is all values of $x$ that make the expression defined.

$(- \infty, 7) \cup (7, \infty)$

Step 11

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 3$

$3 < x < 7$

$x > 7$

Step 12

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.

$\frac{{(- 8)}^{2} + 3 (- 8) - 18}{(- 8) - 7} \geq 0$

The left side $- 1.4\overline{6}$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 6 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{{(0)}^{2} + 3 (0) - 18}{(0) - 7} \geq 0$

The left side $2.\overline{571428}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $3 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $3 < x < 7$ and see if this value makes the original inequality true.

$x = 5$

Replace $x$ with $5$ in the original inequality.

$\frac{{(5)}^{2} + 3 (5) - 18}{(5) - 7} \geq 0$

The left side $- 11$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$\frac{{(10)}^{2} + 3 (10) - 18}{(10) - 7} \geq 0$

The left side $37.\overline{3}$ 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.

$x < - 6$ False

$- 6 < x < 3$ True

$3 < x < 7$ False

$x > 7$ True

$x < - 6$ False

$- 6 < x < 3$ True

$3 < x < 7$ False

$x > 7$ True

Step 13

The solution consists of all of the true intervals.

$- 6 \leq x \leq 3$ or $x > 7$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$- 6 \leq x \leq 3 or x > 7$

Interval Notation:

$[- 6, 3] \cup (7, \infty)$

Step 15

Solve the Inequality for x (x^2+3x-18)/(x-7)>=0

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