Precalculus Examples

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

Step 1

Multiply both sides by $x$ .

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

Step 2

Simplify the left side.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

$1 = 4 x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $4 x = 1$ .

$4 x = 1$

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

Step 4

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

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Step 4.1

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

$x = 0$

Step 4.2

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

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

Step 5

Use each root to create test intervals.

$x < 0$

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

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

Step 6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 6.1.1

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

$x = - 2$

Step 6.1.2

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

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

Step 6.1.3

The left side $- 0.5$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 6.2

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

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Step 6.2.1

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

$x = 0.12$

Step 6.2.2

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

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

Step 6.2.3

The left side $8.\overline{3}$ is not less than the right side $4$ , which means that the given statement is false.

False

Step 6.3

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

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Step 6.3.1

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

$x = 3$

Step 6.3.2

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

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

Step 6.3.3

The left side $0.\overline{3}$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 6.4

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

Step 7

The solution consists of all of the true intervals.

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

Step 8

Convert the inequality to interval notation.

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

Step 9