Precalculus Examples

$| x - \frac{1}{15} | < \frac{1}{5}$

Step 1

Write $| x - \frac{1}{15} | < \frac{1}{5}$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x - \frac{1}{15} \geq 0$

Step 1.2

Add $\frac{1}{15}$ to both sides of the inequality.

$x \geq \frac{1}{15}$

Step 1.3

In the piece where $x - \frac{1}{15}$ is non-negative, remove the absolute value.

$x - \frac{1}{15} < \frac{1}{5}$

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x - \frac{1}{15} < 0$

Step 1.5

Add $\frac{1}{15}$ to both sides of the inequality.

$x < \frac{1}{15}$

Step 1.6

In the piece where $x - \frac{1}{15}$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - \frac{1}{15}) < \frac{1}{5}$

Step 1.7

Write as a piecewise.

${\begin{cases} x - \frac{1}{15} < \frac{1}{5} & x \geq \frac{1}{15} \\ - (x - \frac{1}{15}) < \frac{1}{5} & x < \frac{1}{15} \end{cases}$

Step 1.8

Simplify $- (x - \frac{1}{15}) < \frac{1}{5}$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} x - \frac{1}{15} < \frac{1}{5} & x \geq \frac{1}{15} \\ - x - - \frac{1}{15} < \frac{1}{5} & x < \frac{1}{15} \end{cases}$

Step 1.8.2

Multiply $- - \frac{1}{15}$ .

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Step 1.8.2.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} x - \frac{1}{15} < \frac{1}{5} & x \geq \frac{1}{15} \\ - x + 1 (\frac{1}{15}) < \frac{1}{5} & x < \frac{1}{15} \end{cases}$

Step 1.8.2.2

Multiply $\frac{1}{15}$ by $1$ .

${\begin{cases} x - \frac{1}{15} < \frac{1}{5} & x \geq \frac{1}{15} \\ - x + \frac{1}{15} < \frac{1}{5} & x < \frac{1}{15} \end{cases}$

Step 2

Solve $x - \frac{1}{15} < \frac{1}{5}$ when $x \geq \frac{1}{15}$ .

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Step 2.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1.1

Add $\frac{1}{15}$ to both sides of the inequality.

$x < \frac{1}{5} + \frac{1}{15}$

Step 2.1.2

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x < \frac{1}{5} \cdot \frac{3}{3} + \frac{1}{15}$

Step 2.1.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.3.1

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$x < \frac{3}{5 \cdot 3} + \frac{1}{15}$

Step 2.1.3.2

Multiply $5$ by $3$ .

$x < \frac{3}{15} + \frac{1}{15}$

Step 2.1.4

Combine the numerators over the common denominator.

$x < \frac{3 + 1}{15}$

Step 2.1.5

Add $3$ and $1$ .

$x < \frac{4}{15}$

Step 2.2

Find the intersection of $x < \frac{4}{15}$ and $x \geq \frac{1}{15}$ .

$\frac{1}{15} \leq x < \frac{4}{15}$

Step 3

Solve $- x + \frac{1}{15} < \frac{1}{5}$ when $x < \frac{1}{15}$ .

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Step 3.1

Solve $- x + \frac{1}{15} < \frac{1}{5}$ for $x$ .

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Step 3.1.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.1.1.1

Subtract $\frac{1}{15}$ from both sides of the inequality.

$- x < \frac{1}{5} - \frac{1}{15}$

Step 3.1.1.2

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- x < \frac{1}{5} \cdot \frac{3}{3} - \frac{1}{15}$

Step 3.1.1.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.1.3.1

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$- x < \frac{3}{5 \cdot 3} - \frac{1}{15}$

Step 3.1.1.3.2

Multiply $5$ by $3$ .

$- x < \frac{3}{15} - \frac{1}{15}$

Step 3.1.1.4

Combine the numerators over the common denominator.

$- x < \frac{3 - 1}{15}$

Step 3.1.1.5

Subtract $1$ from $3$ .

$- x < \frac{2}{15}$

Step 3.1.2

Divide each term in $- x < \frac{2}{15}$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- x < \frac{2}{15}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{\frac{2}{15}}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{\frac{2}{15}}{- 1}$

Step 3.1.2.2.2

Divide $x$ by $1$ .

$x > \frac{\frac{2}{15}}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative one from the denominator of $\frac{\frac{2}{15}}{- 1}$ .

$x > - 1 \cdot \frac{2}{15}$

Step 3.1.2.3.2

Rewrite $- 1 \cdot \frac{2}{15}$ as $- \frac{2}{15}$ .

$x > - \frac{2}{15}$

Step 3.2

Find the intersection of $x > - \frac{2}{15}$ and $x < \frac{1}{15}$ .

$- \frac{2}{15} < x < \frac{1}{15}$

Step 4

Find the union of the solutions.

$- \frac{2}{15} < x < \frac{4}{15}$

Step 5

Convert the inequality to interval notation.

$(- \frac{2}{15}, \frac{4}{15})$

Step 6