Precalculus Examples

$| 2 x + 4 |$

Step 1

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

$2 x + 4 \geq 0$

Step 2

Solve the inequality.

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Step 2.1

Subtract $4$ from both sides of the inequality.

$2 x \geq - 4$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 4$ by $2$ .

$x \geq - 2$

Step 3

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

$2 x + 4$

Step 4

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

$2 x + 4 < 0$

Step 5

Solve the inequality.

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Step 5.1

Subtract $4$ from both sides of the inequality.

$2 x < - 4$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 4$ by $2$ .

$x < - 2$

Step 6

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

$- (2 x + 4)$

Step 7

Write as a piecewise.

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

Step 8

Simplify $- (2 x + 4)$ .

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $2$ by $- 1$ .

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

Step 8.3

Multiply $- 1$ by $4$ .

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

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