Algebra Examples

$| 2 x + 4 |$

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

$2 x + 4 \geq 0$

Solve the inequality.

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Subtract $4$ from both sides of the inequality.

$2 x \geq - 4$

Divide each term by $2$ and simplify.

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Divide each term in $2 x \geq - 4$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 4}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 4}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 4}{2}$

Divide $- 4$ by $2$ .

$x \geq - 2$

In the piece where $2 x + 4$ is non-negative, remove the absolute value.

$2 x + 4$

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

$2 x + 4 < 0$

Solve the inequality.

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Subtract $4$ from both sides of the inequality.

$2 x < - 4$

Divide each term by $2$ and simplify.

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Divide each term in $2 x < - 4$ by $2$ .

$\frac{2 x}{2} < \frac{- 4}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} < \frac{- 4}{2}$

Divide $x$ by $1$ .

$x < \frac{- 4}{2}$

Divide $- 4$ by $2$ .

$x < - 2$

In the piece where $2 x + 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x + 4)$

Write as a piecewise.

${\begin{cases} 2 x + 4 & x \geq - 2 \\ - (2 x + 4) & x < - 2 \end{cases}$

Simplify $- (2 x + 4)$ .

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Apply the distributive property.

${\begin{cases} 2 x + 4 & x \geq - 2 \\ - (2 x) - 1 \cdot 4 & x < - 2 \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 4 & x \geq - 2 \\ - 2 x - 1 \cdot 4 & x < - 2 \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} 2 x + 4 & x \geq - 2 \\ - 2 x - 4 & x < - 2 \end{cases}$

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