Algebra Examples

$| 4 m - 4 | < 36$

Step 1

Write $| 4 m - 4 | < 36$ 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.

$4 m - 4 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $4$ to both sides of the inequality.

$4 m \geq 4$

Step 1.2.2

Divide each term in $4 m \geq 4$ by $4$ and simplify.

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Step 1.2.2.1

Divide each term in $4 m \geq 4$ by $4$ .

$\frac{4 m}{4} \geq \frac{4}{4}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 m}{4} \geq \frac{4}{4}$

Step 1.2.2.2.1.2

Divide $m$ by $1$ .

$m \geq \frac{4}{4}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $4$ by $4$ .

$m \geq 1$

Step 1.3

In the piece where $4 m - 4$ is non-negative, remove the absolute value.

$4 m - 4 < 36$

Step 1.4

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

$4 m - 4 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $4$ to both sides of the inequality.

$4 m < 4$

Step 1.5.2

Divide each term in $4 m < 4$ by $4$ and simplify.

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Step 1.5.2.1

Divide each term in $4 m < 4$ by $4$ .

$\frac{4 m}{4} < \frac{4}{4}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $4$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{4 m}{4} < \frac{4}{4}$

Step 1.5.2.2.1.2

Divide $m$ by $1$ .

$m < \frac{4}{4}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $4$ by $4$ .

$m < 1$

Step 1.6

In the piece where $4 m - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 m - 4) < 36$

Step 1.7

Write as a piecewise.

${\begin{cases} 4 m - 4 < 36 & m \geq 1 \\ - (4 m - 4) < 36 & m < 1 \end{cases}$

Step 1.8

Simplify $- (4 m - 4) < 36$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} 4 m - 4 < 36 & m \geq 1 \\ - (4 m) - - 4 < 36 & m < 1 \end{cases}$

Step 1.8.2

Multiply $4$ by $- 1$ .

${\begin{cases} 4 m - 4 < 36 & m \geq 1 \\ - 4 m - - 4 < 36 & m < 1 \end{cases}$

Step 1.8.3

Multiply $- 1$ by $- 4$ .

${\begin{cases} 4 m - 4 < 36 & m \geq 1 \\ - 4 m + 4 < 36 & m < 1 \end{cases}$

Step 2

Solve $4 m - 4 < 36$ when $m \geq 1$ .

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Step 2.1

Solve $4 m - 4 < 36$ for $m$ .

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Step 2.1.1

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

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Step 2.1.1.1

Add $4$ to both sides of the inequality.

$4 m < 36 + 4$

Step 2.1.1.2

Add $36$ and $4$ .

$4 m < 40$

Step 2.1.2

Divide each term in $4 m < 40$ by $4$ and simplify.

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Step 2.1.2.1

Divide each term in $4 m < 40$ by $4$ .

$\frac{4 m}{4} < \frac{40}{4}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{4 m}{4} < \frac{40}{4}$

Step 2.1.2.2.1.2

Divide $m$ by $1$ .

$m < \frac{40}{4}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $40$ by $4$ .

$m < 10$

Step 2.2

Find the intersection of $m < 10$ and $m \geq 1$ .

$1 \leq m < 10$

Step 3

Solve $- 4 m + 4 < 36$ when $m < 1$ .

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Step 3.1

Solve $- 4 m + 4 < 36$ for $m$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $4$ from both sides of the inequality.

$- 4 m < 36 - 4$

Step 3.1.1.2

Subtract $4$ from $36$ .

$- 4 m < 32$

Step 3.1.2

Divide each term in $- 4 m < 32$ by $- 4$ and simplify.

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Step 3.1.2.1

Divide each term in $- 4 m < 32$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 m}{- 4} > \frac{32}{- 4}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 m}{- 4} > \frac{32}{- 4}$

Step 3.1.2.2.1.2

Divide $m$ by $1$ .

$m > \frac{32}{- 4}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $32$ by $- 4$ .

$m > - 8$

Step 3.2

Find the intersection of $m > - 8$ and $m < 1$ .

$- 8 < m < 1$

Step 4

Find the union of the solutions.

$- 8 < m < 10$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 8 < m < 10$

Interval Notation:

$(- 8, 10)$

Step 6