Examples

$| 5 x - 5 |$

Step 1

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

$5 x - 5 \geq 0$

Step 2

Solve the inequality.

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Step 2.1

Add $5$ to both sides of the inequality.

$5 x \geq 5$

Step 2.2

Divide each term in $5 x \geq 5$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 x \geq 5$ by $5$ .

$\frac{5 x}{5} \geq \frac{5}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{5}{5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{5}{5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $5$ by $5$ .

$x \geq 1$

Step 3

In the piece where $5 x - 5$ is non-negative, remove the absolute value.

$5 x - 5$

Step 4

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

$5 x - 5 < 0$

Step 5

Solve the inequality.

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Step 5.1

Add $5$ to both sides of the inequality.

$5 x < 5$

Step 5.2

Divide each term in $5 x < 5$ by $5$ and simplify.

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Step 5.2.1

Divide each term in $5 x < 5$ by $5$ .

$\frac{5 x}{5} < \frac{5}{5}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} < \frac{5}{5}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{5}{5}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $5$ by $5$ .

$x < 1$

Step 6

In the piece where $5 x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 x - 5)$

Step 7

Write as a piecewise.

${\begin{cases} 5 x - 5 & x \geq 1 \\ - (5 x - 5) & x < 1 \end{cases}$

Step 8

Simplify $- (5 x - 5)$ .

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Step 8.1

Apply the distributive property.

${\begin{cases} 5 x - 5 & x \geq 1 \\ - (5 x) - - 5 & x < 1 \end{cases}$

Step 8.2

Multiply $5$ by $- 1$ .

${\begin{cases} 5 x - 5 & x \geq 1 \\ - 5 x - - 5 & x < 1 \end{cases}$

Step 8.3

Multiply $- 1$ by $- 5$ .

${\begin{cases} 5 x - 5 & x \geq 1 \\ - 5 x + 5 & x < 1 \end{cases}$

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