Algebra Examples

$| \frac{m - 15}{5} | < 16$

Step 1

Write $| \frac{m - 15}{5} | < 16$ 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.

$\frac{m - 15}{5} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $5$ .

$\frac{m - 15}{5} \cdot 5 \geq 0 \cdot 5$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$\frac{m - 15}{5} \cdot 5 \geq 0 \cdot 5$

Step 1.2.2.1.1.2

Rewrite the expression.

$m - 15 \geq 0 \cdot 5$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $5$ .

$m - 15 \geq 0$

Step 1.2.3

Add $15$ to both sides of the inequality.

$m \geq 15$

Step 1.3

In the piece where $\frac{m - 15}{5}$ is non-negative, remove the absolute value.

$\frac{m - 15}{5} < 16$

Step 1.4

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

$\frac{m - 15}{5} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $5$ .

$\frac{m - 15}{5} \cdot 5 < 0 \cdot 5$

Step 1.5.2

Simplify.

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Step 1.5.2.1

Simplify the left side.

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Step 1.5.2.1.1

Cancel the common factor of $5$ .

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Step 1.5.2.1.1.1

Cancel the common factor.

$\frac{m - 15}{5} \cdot 5 < 0 \cdot 5$

Step 1.5.2.1.1.2

Rewrite the expression.

$m - 15 < 0 \cdot 5$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $5$ .

$m - 15 < 0$

Step 1.5.3

Add $15$ to both sides of the inequality.

$m < 15$

Step 1.6

In the piece where $\frac{m - 15}{5}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{m - 15}{5} < 16$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{m - 15}{5} < 16 & m \geq 15 \\ - \frac{m - 15}{5} < 16 & m < 15 \end{cases}$

Step 2

Solve $\frac{m - 15}{5} < 16$ when $m \geq 15$ .

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Step 2.1

Solve $\frac{m - 15}{5} < 16$ for $m$ .

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Step 2.1.1

Multiply both sides by $5$ .

$\frac{m - 15}{5} \cdot 5 < 16 \cdot 5$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify the left side.

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Step 2.1.2.1.1

Cancel the common factor of $5$ .

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Step 2.1.2.1.1.1

Cancel the common factor.

$\frac{m - 15}{5} \cdot 5 < 16 \cdot 5$

Step 2.1.2.1.1.2

Rewrite the expression.

$m - 15 < 16 \cdot 5$

Step 2.1.2.2

Simplify the right side.

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Step 2.1.2.2.1

Multiply $16$ by $5$ .

$m - 15 < 80$

Step 2.1.3

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

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Step 2.1.3.1

Add $15$ to both sides of the inequality.

$m < 80 + 15$

Step 2.1.3.2

Add $80$ and $15$ .

$m < 95$

Step 2.2

Find the intersection of $m < 95$ and $m \geq 15$ .

$15 \leq m < 95$

Step 3

Solve $- \frac{m - 15}{5} < 16$ when $m < 15$ .

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Step 3.1

Solve $- \frac{m - 15}{5} < 16$ for $m$ .

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Step 3.1.1

Divide each term in $- \frac{m - 15}{5} < 16$ by $- 1$ and simplify.

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Step 3.1.1.1

Divide each term in $- \frac{m - 15}{5} < 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{m - 15}{5}}{- 1} > \frac{16}{- 1}$

Step 3.1.1.2

Simplify the left side.

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Step 3.1.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{m - 15}{5}}{1} > \frac{16}{- 1}$

Step 3.1.1.2.2

Divide $\frac{m - 15}{5}$ by $1$ .

$\frac{m - 15}{5} > \frac{16}{- 1}$

Step 3.1.1.3

Simplify the right side.

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Step 3.1.1.3.1

Divide $16$ by $- 1$ .

$\frac{m - 15}{5} > - 16$

Step 3.1.2

Multiply both sides by $5$ .

$\frac{m - 15}{5} \cdot 5 > - 16 \cdot 5$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Simplify the left side.

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Step 3.1.3.1.1

Cancel the common factor of $5$ .

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Step 3.1.3.1.1.1

Cancel the common factor.

$\frac{m - 15}{5} \cdot 5 > - 16 \cdot 5$

Step 3.1.3.1.1.2

Rewrite the expression.

$m - 15 > - 16 \cdot 5$

Step 3.1.3.2

Simplify the right side.

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Step 3.1.3.2.1

Multiply $- 16$ by $5$ .

$m - 15 > - 80$

Step 3.1.4

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

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Step 3.1.4.1

Add $15$ to both sides of the inequality.

$m > - 80 + 15$

Step 3.1.4.2

Add $- 80$ and $15$ .

$m > - 65$

Step 3.2

Find the intersection of $m > - 65$ and $m < 15$ .

$- 65 < m < 15$

Step 4

Find the union of the solutions.

$- 65 < m < 95$

Step 5

Convert the inequality to interval notation.

$(- 65, 95)$

Step 6