Finite Math Examples

$| \frac{1}{8} - x | < 3$

Step 1

Write $| \frac{1}{8} - x | < 3$ 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.

$\frac{1}{8} - x \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $\frac{1}{8}$ from both sides of the inequality.

$- x \geq - \frac{1}{8}$

Step 1.2.2

Divide each term in $- x \geq - \frac{1}{8}$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x \geq - \frac{1}{8}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- \frac{1}{8}}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- \frac{1}{8}}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- \frac{1}{8}}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{\frac{1}{8}}{1}$

Step 1.2.2.3.2

Divide $\frac{1}{8}$ by $1$ .

$x \leq \frac{1}{8}$

Step 1.3

In the piece where $\frac{1}{8} - x$ is non-negative, remove the absolute value.

$\frac{1}{8} - x < 3$

Step 1.4

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

$\frac{1}{8} - x < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $\frac{1}{8}$ from both sides of the inequality.

$- x < - \frac{1}{8}$

Step 1.5.2

Divide each term in $- x < - \frac{1}{8}$ by $- 1$ and simplify.

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Step 1.5.2.1

Divide each term in $- x < - \frac{1}{8}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- \frac{1}{8}}{- 1}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- \frac{1}{8}}{- 1}$

Step 1.5.2.2.2

Divide $x$ by $1$ .

$x > \frac{- \frac{1}{8}}{- 1}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Dividing two negative values results in a positive value.

$x > \frac{\frac{1}{8}}{1}$

Step 1.5.2.3.2

Divide $\frac{1}{8}$ by $1$ .

$x > \frac{1}{8}$

Step 1.6

In the piece where $\frac{1}{8} - x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (\frac{1}{8} - x) < 3$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{1}{8} - x < 3 & x \leq \frac{1}{8} \\ - (\frac{1}{8} - x) < 3 & x > \frac{1}{8} \end{cases}$

Step 1.8

Simplify $- (\frac{1}{8} - x) < 3$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} \frac{1}{8} - x < 3 & x \leq \frac{1}{8} \\ - \frac{1}{8} - - x < 3 & x > \frac{1}{8} \end{cases}$

Step 1.8.2

Multiply $- - x$ .

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Step 1.8.2.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} \frac{1}{8} - x < 3 & x \leq \frac{1}{8} \\ - \frac{1}{8} + 1 x < 3 & x > \frac{1}{8} \end{cases}$

Step 1.8.2.2

Multiply $x$ by $1$ .

${\begin{cases} \frac{1}{8} - x < 3 & x \leq \frac{1}{8} \\ - \frac{1}{8} + x < 3 & x > \frac{1}{8} \end{cases}$

Step 2

Solve $\frac{1}{8} - x < 3$ when $x \leq \frac{1}{8}$ .

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Step 2.1

Solve $\frac{1}{8} - x < 3$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $\frac{1}{8}$ from both sides of the inequality.

$- x < 3 - \frac{1}{8}$

Step 2.1.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- x < 3 \cdot \frac{8}{8} - \frac{1}{8}$

Step 2.1.1.3

Combine $3$ and $\frac{8}{8}$ .

$- x < \frac{3 \cdot 8}{8} - \frac{1}{8}$

Step 2.1.1.4

Combine the numerators over the common denominator.

$- x < \frac{3 \cdot 8 - 1}{8}$

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

Multiply $3$ by $8$ .

$- x < \frac{24 - 1}{8}$

Step 2.1.1.5.2

Subtract $1$ from $24$ .

$- x < \frac{23}{8}$

Step 2.1.2

Divide each term in $- x < \frac{23}{8}$ by $- 1$ and simplify.

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Step 2.1.2.1

Divide each term in $- x < \frac{23}{8}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{\frac{23}{8}}{- 1}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{\frac{23}{8}}{- 1}$

Step 2.1.2.2.2

Divide $x$ by $1$ .

$x > \frac{\frac{23}{8}}{- 1}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative one from the denominator of $\frac{\frac{23}{8}}{- 1}$ .

$x > - 1 \cdot \frac{23}{8}$

Step 2.1.2.3.2

Rewrite $- 1 \cdot \frac{23}{8}$ as $- \frac{23}{8}$ .

$x > - \frac{23}{8}$

Step 2.2

Find the intersection of $x > - \frac{23}{8}$ and $x \leq \frac{1}{8}$ .

$- \frac{23}{8} < x \leq \frac{1}{8}$

Step 3

Solve $- \frac{1}{8} + x < 3$ when $x > \frac{1}{8}$ .

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Step 3.1

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

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Step 3.1.1

Add $\frac{1}{8}$ to both sides of the inequality.

$x < 3 + \frac{1}{8}$

Step 3.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x < 3 \cdot \frac{8}{8} + \frac{1}{8}$

Step 3.1.3

Combine $3$ and $\frac{8}{8}$ .

$x < \frac{3 \cdot 8}{8} + \frac{1}{8}$

Step 3.1.4

Combine the numerators over the common denominator.

$x < \frac{3 \cdot 8 + 1}{8}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $3$ by $8$ .

$x < \frac{24 + 1}{8}$

Step 3.1.5.2

Add $24$ and $1$ .

$x < \frac{25}{8}$

Step 3.2

Find the intersection of $x < \frac{25}{8}$ and $x > \frac{1}{8}$ .

$\frac{1}{8} < x < \frac{25}{8}$

Step 4

Find the union of the solutions.

$- \frac{23}{8} < x < \frac{25}{8}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{23}{8} < x < \frac{25}{8}$

Interval Notation:

$(- \frac{23}{8}, \frac{25}{8})$

Step 6