Solve the Inequality for x (5x+1)/(x-4)<=0

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

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

$5 x + 1 = 0$

$x - 4 = 0$

Subtract $1$ from both sides of the equation.

$5 x = - 1$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Add $4$ to both sides of the equation.

$x = 4$

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

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

$x = 4$

Consolidate the solutions.

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

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

Use each root to create test intervals.

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

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

$x > 4$

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 $x < - \frac{1}{5}$ to see if it makes the inequality true.

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

$x = - 3$

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

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

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

False

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

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

$x = 0$

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

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

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

True

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.

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

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

False

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

The solution consists of all of the true intervals.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Solve the Inequality for x (5x+1)/(x-4)<=0

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