Precalculus Examples

$| 4 x - 12 |$

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

$4 x - 12 \geq 0$

Solve the inequality.

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Add $12$ to both sides of the inequality.

$4 x \geq 12$

Divide each term by $4$ and simplify.

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

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Divide $12$ by $4$ .

$x \geq 3$

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

$4 x - 12$

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

$4 x - 12 < 0$

Solve the inequality.

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Add $12$ to both sides of the inequality.

$4 x < 12$

Divide each term by $4$ and simplify.

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

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Divide $12$ by $4$ .

$x < 3$

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

$- (4 x - 12)$

Write as a piecewise.

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

Simplify $- (4 x - 12)$ .

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Apply the distributive property.

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

Multiply $4$ by $- 1$ .

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

Multiply $- 1$ by $- 12$ .

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

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