Pre-Algebra Examples

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

Step 1

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

$x - 8 = 0$

$x + 5 = 0$

Step 2

Add $8$ to both sides of the equation.

$x = 8$

Step 3

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4

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

$x = 8$

$x = - 5$

Step 5

Consolidate the solutions.

$x = 8, - 5$

Step 6

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

Tap for more steps...

Step 6.1

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

$x + 5 = 0$

Step 6.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6.3

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

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

Step 7

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 8$

$x > 8$

Step 8

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 8.1

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

Tap for more steps...

Step 8.1.1

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

$x = - 8$

Step 8.1.2

Replace $x$ with $- 8$ in the original inequality.

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

Step 8.1.3

The left side $5.\overline{3}$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.2

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

Tap for more steps...

Step 8.2.1

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

$x = 0$

Step 8.2.2

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

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

Step 8.2.3

The left side $- 1.6$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

$x = 10$

Step 8.3.2

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

$\frac{(10) - 8}{(10) + 5} < 0$

Step 8.3.3

The left side $0.1\overline{3}$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < - 5$ False

$- 5 < x < 8$ True

$x > 8$ False

$x < - 5$ False

$- 5 < x < 8$ True

$x > 8$ False

Step 9

The solution consists of all of the true intervals.

$- 5 < x < 8$

Step 10