Algebra Examples

$| - 12 + 4 x |$

Step 1

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

$- 12 + 4 x \geq 0$

Step 2

Solve the inequality.

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Step 2.1

Add $12$ to both sides of the inequality.

$4 x \geq 12$

Step 2.2

Divide each term in $4 x \geq 12$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 x \geq 12$ by $4$ .

$\frac{4 x}{4} \geq \frac{12}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{12}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{12}{4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $12$ by $4$ .

$x \geq 3$

Step 3

In the piece where $- 12 + 4 x$ is non-negative, remove the absolute value.

$- 12 + 4 x$

Step 4

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

$- 12 + 4 x < 0$

Step 5

Solve the inequality.

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Step 5.1

Add $12$ to both sides of the inequality.

$4 x < 12$

Step 5.2

Divide each term in $4 x < 12$ by $4$ and simplify.

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Step 5.2.1

Divide each term in $4 x < 12$ by $4$ .

$\frac{4 x}{4} < \frac{12}{4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{12}{4}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{12}{4}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $12$ by $4$ .

$x < 3$

Step 6

In the piece where $- 12 + 4 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 12 + 4 x)$

Step 7

Write as a piecewise.

${\begin{cases} - 12 + 4 x & x \geq 3 \\ - (- 12 + 4 x) & x < 3 \end{cases}$

Step 8

Simplify $- (- 12 + 4 x)$ .

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Step 8.1

Apply the distributive property.

${\begin{cases} - 12 + 4 x & x \geq 3 \\ - - 12 - (4 x) & x < 3 \end{cases}$

Step 8.2

Multiply $- 1$ by $- 12$ .

${\begin{cases} - 12 + 4 x & x \geq 3 \\ 12 - (4 x) & x < 3 \end{cases}$

Step 8.3

Multiply $4$ by $- 1$ .

${\begin{cases} - 12 + 4 x & x \geq 3 \\ 12 - 4 x & x < 3 \end{cases}$