Trigonometry Examples

$\frac{x - 4}{x + 5} < 4$

Step 1

Subtract $4$ from both sides of the inequality.

$\frac{x - 4}{x + 5} - 4 < 0$

Step 2

Simplify $\frac{x - 4}{x + 5} - 4$ .

Tap for more steps...

Step 2.1

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{x - 4}{x + 5} - 4 \cdot \frac{x + 5}{x + 5} < 0$

Step 2.2

Combine $- 4$ and $\frac{x + 5}{x + 5}$ .

$\frac{x - 4}{x + 5} + \frac{- 4 (x + 5)}{x + 5} < 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x - 4 - 4 (x + 5)}{x + 5} < 0$

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$\frac{x - 4 - 4 x - 4 \cdot 5}{x + 5} < 0$

Step 2.4.2

Multiply $- 4$ by $5$ .

$\frac{x - 4 - 4 x - 20}{x + 5} < 0$

Step 2.4.3

Subtract $4 x$ from $x$ .

$\frac{- 3 x - 4 - 20}{x + 5} < 0$

Step 2.4.4

Subtract $20$ from $- 4$ .

$\frac{- 3 x - 24}{x + 5} < 0$

Step 2.4.5

Factor $3$ out of $- 3 x - 24$ .

Tap for more steps...

Step 2.4.5.1

Factor $3$ out of $- 3 x$ .

$\frac{3 (- x) - 24}{x + 5} < 0$

Step 2.4.5.2

Factor $3$ out of $- 24$ .

$\frac{3 (- x) + 3 (- 8)}{x + 5} < 0$

Step 2.4.5.3

Factor $3$ out of $3 (- x) + 3 (- 8)$ .

$\frac{3 (- x - 8)}{x + 5} < 0$

Step 2.5

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

$\frac{3 (- (x) - 8)}{x + 5} < 0$

Step 2.6

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{3 (- (x) - 1 (8))}{x + 5} < 0$

Step 2.7

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

$\frac{3 (- (x + 8))}{x + 5} < 0$

Step 2.8

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

$\frac{3 (- 1 (x + 8))}{x + 5} < 0$

Step 2.9

Move the negative in front of the fraction.

$- \frac{3 (x + 8)}{x + 5} < 0$

Step 3

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

$x + 8 = 0$

$x + 5 = 0$

Step 4

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 5

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6

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

$x = - 8$

$x = - 5$

Step 7

Consolidate the solutions.

$x = - 8, - 5$

Step 8

Find the domain of $- \frac{3 (x + 8)}{x + 5}$ .

Tap for more steps...

Step 8.1

Set the denominator in $\frac{3 (x + 8)}{x + 5}$ equal to $0$ to find where the expression is undefined.

$x + 5 = 0$

Step 8.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 8.3

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

$(- \infty, - 5) \cup (- 5, \infty)$

Step 9

Use each root to create test intervals.

$x < - 8$

$- 8 < x < - 5$

$x > - 5$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 10.1

Test a value on the interval $x < - 8$ to see if it makes the inequality true.

Tap for more steps...

Step 10.1.1

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

Step 10.1.2

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

$\frac{(- 10) - 4}{(- 10) + 5} < 4$

Step 10.1.3

The left side $2.8$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- 8 < x < - 5$ to see if it makes the inequality true.

Tap for more steps...

Step 10.2.1

Choose a value on the interval $- 8 < x < - 5$ and see if this value makes the original inequality true.

$x = - 6$

Step 10.2.2

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

$\frac{(- 6) - 4}{(- 6) + 5} < 4$

Step 10.2.3

The left side $10$ is not less than the right side $4$ , which means that the given statement is false.

False

Step 10.3

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

Tap for more steps...

Step 10.3.1

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

$x = 0$

Step 10.3.2

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

$\frac{(0) - 4}{(0) + 5} < 4$

Step 10.3.3

The left side $- 0.8$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 10.4

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

$x < - 8$ True

$- 8 < x < - 5$ False

$x > - 5$ True

$x < - 8$ True

$- 8 < x < - 5$ False

$x > - 5$ True

Step 11

The solution consists of all of the true intervals.

$x < - 8$ or $x > - 5$

Step 12

Convert the inequality to interval notation.

$(- \infty, - 8) \cup (- 5, \infty)$

Step 13