Algebra Examples

$| \frac{m - 12}{4} | < 19$

Step 1

Write $| \frac{m - 12}{4} | < 19$ 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.

$\frac{m - 12}{4} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $4$ .

$\frac{m - 12}{4} \cdot 4 \geq 0 \cdot 4$

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 $4$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$\frac{m - 12}{4} \cdot 4 \geq 0 \cdot 4$

Step 1.2.2.1.1.2

Rewrite the expression.

$m - 12 \geq 0 \cdot 4$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $4$ .

$m - 12 \geq 0$

Step 1.2.3

Add $12$ to both sides of the inequality.

$m \geq 12$

Step 1.3

In the piece where $\frac{m - 12}{4}$ is non-negative, remove the absolute value.

$\frac{m - 12}{4} < 19$

Step 1.4

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

$\frac{m - 12}{4} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $4$ .

$\frac{m - 12}{4} \cdot 4 < 0 \cdot 4$

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 $4$ .

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Step 1.5.2.1.1.1

Cancel the common factor.

$\frac{m - 12}{4} \cdot 4 < 0 \cdot 4$

Step 1.5.2.1.1.2

Rewrite the expression.

$m - 12 < 0 \cdot 4$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $4$ .

$m - 12 < 0$

Step 1.5.3

Add $12$ to both sides of the inequality.

$m < 12$

Step 1.6

In the piece where $\frac{m - 12}{4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{m - 12}{4} < 19$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{m - 12}{4} < 19 & m \geq 12 \\ - \frac{m - 12}{4} < 19 & m < 12 \end{cases}$

Step 2

Solve $\frac{m - 12}{4} < 19$ when $m \geq 12$ .

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Step 2.1

Solve $\frac{m - 12}{4} < 19$ for $m$ .

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Step 2.1.1

Multiply both sides by $4$ .

$\frac{m - 12}{4} \cdot 4 < 19 \cdot 4$

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 $4$ .

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Step 2.1.2.1.1.1

Cancel the common factor.

$\frac{m - 12}{4} \cdot 4 < 19 \cdot 4$

Step 2.1.2.1.1.2

Rewrite the expression.

$m - 12 < 19 \cdot 4$

Step 2.1.2.2

Simplify the right side.

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Step 2.1.2.2.1

Multiply $19$ by $4$ .

$m - 12 < 76$

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 $12$ to both sides of the inequality.

$m < 76 + 12$

Step 2.1.3.2

Add $76$ and $12$ .

$m < 88$

Step 2.2

Find the intersection of $m < 88$ and $m \geq 12$ .

$12 \leq m < 88$

Step 3

Solve $- \frac{m - 12}{4} < 19$ when $m < 12$ .

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Step 3.1

Solve $- \frac{m - 12}{4} < 19$ for $m$ .

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Step 3.1.1

Divide each term in $- \frac{m - 12}{4} < 19$ by $- 1$ and simplify.

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Step 3.1.1.1

Divide each term in $- \frac{m - 12}{4} < 19$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{m - 12}{4}}{- 1} > \frac{19}{- 1}$

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.

$\frac{\frac{m - 12}{4}}{1} > \frac{19}{- 1}$

Step 3.1.1.2.2

Divide $\frac{m - 12}{4}$ by $1$ .

$\frac{m - 12}{4} > \frac{19}{- 1}$

Step 3.1.1.3

Simplify the right side.

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Step 3.1.1.3.1

Divide $19$ by $- 1$ .

$\frac{m - 12}{4} > - 19$

Step 3.1.2

Multiply both sides by $4$ .

$\frac{m - 12}{4} \cdot 4 > - 19 \cdot 4$

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 $4$ .

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Step 3.1.3.1.1.1

Cancel the common factor.

$\frac{m - 12}{4} \cdot 4 > - 19 \cdot 4$

Step 3.1.3.1.1.2

Rewrite the expression.

$m - 12 > - 19 \cdot 4$

Step 3.1.3.2

Simplify the right side.

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Step 3.1.3.2.1

Multiply $- 19$ by $4$ .

$m - 12 > - 76$

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 $12$ to both sides of the inequality.

$m > - 76 + 12$

Step 3.1.4.2

Add $- 76$ and $12$ .

$m > - 64$

Step 3.2

Find the intersection of $m > - 64$ and $m < 12$ .

$- 64 < m < 12$

Step 4

Find the union of the solutions.

$- 64 < m < 88$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 64 < m < 88$

Interval Notation:

$(- 64, 88)$

Step 6