Algebra Examples

$\frac{5 x + 1}{x - 4} \leq 0$

Step 1

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

$5 x + 1 = 0$

$x - 4 = 0$

Step 2

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Step 3

Divide each term in $5 x = - 1$ by $5$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Step 4

Add $4$ to both sides of the equation.

$x = 4$

Step 5

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

$x = - \frac{1}{5}$

$x = 4$

Step 6

Consolidate the solutions.

$x = - \frac{1}{5}, 4$

Step 7

Find the domain of $\frac{5 x + 1}{x - 4}$ .

Tap for more steps...

Step 7.1

Set the denominator in $\frac{5 x + 1}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 7.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7.3

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

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

Step 8

Use each root to create test intervals.

$x < - \frac{1}{5}$

$- \frac{1}{5} < x < 4$

$x > 4$

Step 9

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 9.1

Test a value on the interval $x < - \frac{1}{5}$ to see if it makes the inequality true.

Tap for more steps...

Step 9.1.1

Choose a value on the interval $x < - \frac{1}{5}$ and see if this value makes the original inequality true.

$x = - 3$

Step 9.1.2

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

$\frac{5 (- 3) + 1}{(- 3) - 4} \leq 0$

Step 9.1.3

The left side $2$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- \frac{1}{5} < x < 4$ to see if it makes the inequality true.

Tap for more steps...

Step 9.2.1

Choose a value on the interval $- \frac{1}{5} < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

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

$\frac{5 (0) + 1}{(0) - 4} \leq 0$

Step 9.2.3

The left side $- 0.25$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Tap for more steps...

Step 9.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 9.3.2

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

$\frac{5 (6) + 1}{(6) - 4} \leq 0$

Step 9.3.3

The left side $15.5$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 9.4

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

$x < - \frac{1}{5}$ False

$- \frac{1}{5} < x < 4$ True

$x > 4$ False

$x < - \frac{1}{5}$ False

$- \frac{1}{5} < x < 4$ True

$x > 4$ False

Step 10

The solution consists of all of the true intervals.

$- \frac{1}{5} \leq x < 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- \frac{1}{5} \leq x < 4$

Interval Notation:

$[- \frac{1}{5}, 4)$

Step 12