Solve the Inequality for x (4x-1)/x>0

$\frac{4 x - 1}{x} > 0$

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

$x = 0$

$4 x - 1 = 0$

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

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

$x = 0$

$x = \frac{1}{4}$

Consolidate the solutions.

$x = 0, \frac{1}{4}$

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

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

$x = 0$

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

$(- \infty, 0) \cup (0, \infty)$

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{1}{4}$

$x > \frac{1}{4}$

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 $x < 0$ to see if it makes the inequality true.

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$\frac{4 (- 2) - 1}{- 2} > 0$

The left side $4.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < \frac{1}{4}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{1}{4}$ and see if this value makes the original inequality true.

$x = 0.12$

Replace $x$ with $0.12$ in the original inequality.

$\frac{4 (0.12) - 1}{0.12} > 0$

The left side $- 4.\overline{3}$ is not greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 3$

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

$\frac{4 (3) - 1}{3} > 0$

The left side $3.\overline{6}$ is greater 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.

$x < 0$ True

$0 < x < \frac{1}{4}$ False

$x > \frac{1}{4}$ True

$x < 0$ True

$0 < x < \frac{1}{4}$ False

$x > \frac{1}{4}$ True

The solution consists of all of the true intervals.

$x < 0$ or $x > \frac{1}{4}$

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or x > \frac{1}{4}$

Interval Notation:

$(- \infty, 0) \cup (\frac{1}{4}, \infty)$

Solve the Inequality for x (4x-1)/x>0

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