Solve the Inequality for x (x+27)/(x+4)<9

$\frac{x + 27}{x + 4} < 9$

Step 1

Subtract $9$ from both sides of the inequality.

$\frac{x + 27}{x + 4} - 9 < 0$

Step 2

Simplify $\frac{x + 27}{x + 4} - 9$ .

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{x + 27}{x + 4} - 9 \cdot \frac{x + 4}{x + 4} < 0$

Combine $- 9$ and $\frac{x + 4}{x + 4}$ .

$\frac{x + 27}{x + 4} + \frac{- 9 (x + 4)}{x + 4} < 0$

Combine the numerators over the common denominator.

$\frac{x + 27 - 9 (x + 4)}{x + 4} < 0$

Simplify the numerator.

Apply the distributive property.

$\frac{x + 27 - 9 x - 9 \cdot 4}{x + 4} < 0$

Multiply $- 9$ by $4$ .

$\frac{x + 27 - 9 x - 36}{x + 4} < 0$

Subtract $9 x$ from $x$ .

$\frac{- 8 x + 27 - 36}{x + 4} < 0$

Subtract $36$ from $27$ .

$\frac{- 8 x - 9}{x + 4} < 0$

Factor $- 1$ out of $- 8 x$ .

$\frac{- (8 x) - 9}{x + 4} < 0$

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (8 x) - 1 (9)}{x + 4} < 0$

Factor $- 1$ out of $- (8 x) - 1 (9)$ .

$\frac{- (8 x + 9)}{x + 4} < 0$

Rewrite $- (8 x + 9)$ as $- 1 (8 x + 9)$ .

$\frac{- 1 (8 x + 9)}{x + 4} < 0$

Move the negative in front of the fraction.

$- \frac{8 x + 9}{x + 4} < 0$

Step 3

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

$8 x + 9 = 0$

$x + 4 = 0$

Step 4

Subtract $9$ from both sides of the equation.

$8 x = - 9$

Step 5

Divide each term in $8 x = - 9$ by $8$ and simplify.

Divide each term in $8 x = - 9$ by $8$ .

$\frac{8 x}{8} = \frac{- 9}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 9}{8}$

Divide $x$ by $1$ .

$x = \frac{- 9}{8}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{9}{8}$

Step 6

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

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

$x = - \frac{9}{8}$

$x = - 4$

Step 8

Consolidate the solutions.

$x = - \frac{9}{8}, - 4$

Step 9

Find the domain of $- \frac{8 x + 9}{x + 4}$ .

Set the denominator in $\frac{8 x + 9}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

$(- \infty, - 4) \cup (- 4, \infty)$

Step 10

Use each root to create test intervals.

$x < - 4$

$- 4 < x < - \frac{9}{8}$

$x > - \frac{9}{8}$

Step 11

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 $x < - 4$ 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.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$\frac{(- 6) + 27}{(- 6) + 4} < 9$

The left side $- 10.5$ is less than the right side $9$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < - \frac{9}{8}$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < - \frac{9}{8}$ and see if this value makes the original inequality true.

$x = - 3$

Replace $x$ with $- 3$ in the original inequality.

$\frac{(- 3) + 27}{(- 3) + 4} < 9$

The left side $24$ is not less than the right side $9$ , which means that the given statement is false.

False

Test a value on the interval $x > - \frac{9}{8}$ to see if it makes the inequality true.

Choose a value on the interval $x > - \frac{9}{8}$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{(0) + 27}{(0) + 4} < 9$

The left side $6.75$ is less than the right side $9$ , which means that the given statement is always true.

True

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

$x < - 4$ True

$- 4 < x < - \frac{9}{8}$ False

$x > - \frac{9}{8}$ True

$x < - 4$ True

$- 4 < x < - \frac{9}{8}$ False

$x > - \frac{9}{8}$ True

Step 12

The solution consists of all of the true intervals.

$x < - 4$ or $x > - \frac{9}{8}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x < - 4 or x > - \frac{9}{8}$

Interval Notation:

$(- \infty, - 4) \cup (- \frac{9}{8}, \infty)$

Step 14

Solve the Inequality for x (x+27)/(x+4)<9

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