Solve the Inequality for x 1/x<4

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

Multiply both sides by $x$ .

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

$1 = 4 x$

Solve for $x$ .

Rewrite the equation as $4 x = 1$ .

$4 x = 1$

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 $1$ .

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

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

Set the denominator in $\frac{1}{x}$ 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.

$\frac{1}{- 2} < 4$

The left side $- 0.5$ is less than the right side $4$ , 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.

$\frac{1}{0.12} < 4$

The left side $8.\overline{3}$ is not less than the right side $4$ , 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.

$\frac{1}{3} < 4$

The left side $0.\overline{3}$ is less than the right side $4$ , 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 1/x<4

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