Solve the Inequality for x (3x-1)/(1-5x)<=0

$\frac{3 x - 1}{1 - 5 x} \leq 0$

Step 1

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

$3 x - 1 = 0$

$1 - 5 x = 0$

Step 2

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 4

Subtract $1$ from both sides of the equation.

$- 5 x = - 1$

Step 5

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.

Dividing two negative values results in a positive value.

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

Step 6

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

$x = \frac{1}{3}$

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

Step 7

Consolidate the solutions.

$x = \frac{1}{3}, \frac{1}{5}$

Step 8

Find the domain of $\frac{3 x - 1}{1 - 5 x}$ .

Set the denominator in $\frac{3 x - 1}{1 - 5 x}$ equal to $0$ to find where the expression is undefined.

$1 - 5 x = 0$

Solve for $x$ .

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.

Dividing two negative values results in a positive value.

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

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

$(- \infty, \frac{1}{5}) \cup (\frac{1}{5}, \infty)$

Step 9

Use each root to create test intervals.

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

$\frac{1}{5} < x < \frac{1}{3}$

$x > \frac{1}{3}$

Step 10

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.

$x = 0$

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

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

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $\frac{1}{5} < x < \frac{1}{3}$ to see if it makes the inequality true.

Choose a value on the interval $\frac{1}{5} < x < \frac{1}{3}$ and see if this value makes the original inequality true.

$x = 0.27$

Replace $x$ with $0.27$ in the original inequality.

$\frac{3 (0.27) - 1}{1 - 5 \cdot 0.27} \leq 0$

The left side $0.5\overline{428571}$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{1}{3}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{1}{3}$ and see if this value makes the original inequality true.

$x = 3$

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

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

The left side $- 0.\overline{571428}$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$\frac{1}{5} < x < \frac{1}{3}$ False

$x > \frac{1}{3}$ True

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

$\frac{1}{5} < x < \frac{1}{3}$ False

$x > \frac{1}{3}$ True

Step 11

The solution consists of all of the true intervals.

$x < \frac{1}{5}$ or $x \geq \frac{1}{3}$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{1}{5} or x \geq \frac{1}{3}$

Interval Notation:

$(- \infty, \frac{1}{5}) \cup [\frac{1}{3}, \infty)$

Step 13

Solve the Inequality for x (3x-1)/(1-5x)<=0

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