Write the Absolute Value as Piecewise f(x)=|2-2x|

$f (x) = | 2 - 2 x |$

Step 1

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

$2 - 2 x \geq 0$

Step 2

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$- 2 x \geq - 2$

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

Divide each term in $- 2 x \geq - 2$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 2}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 2}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{- 2}{- 2}$

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x \leq 1$

Step 3

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

$2 - 2 x$

Step 4

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

$2 - 2 x < 0$

Step 5

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$- 2 x < - 2$

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

Divide each term in $- 2 x < - 2$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{- 2}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{- 2}{- 2}$

Divide $x$ by $1$ .

$x > \frac{- 2}{- 2}$

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x > 1$

Step 6

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

$- (2 - 2 x)$

Step 7

Write as a piecewise.

$f (x) = {\begin{cases} 2 - 2 x & x \leq 1 \\ - (2 - 2 x) & x > 1 \end{cases}$

Step 8

Simplify $- (2 - 2 x)$ .

Apply the distributive property.

$f (x) = {\begin{cases} 2 - 2 x & x \leq 1 \\ - 1 \cdot 2 - (- 2 x) & x > 1 \end{cases}$

Multiply $- 1$ by $2$ .

$f (x) = {\begin{cases} 2 - 2 x & x \leq 1 \\ - 2 - (- 2 x) & x > 1 \end{cases}$

Multiply $- 2$ by $- 1$ .

$f (x) = {\begin{cases} 2 - 2 x & x \leq 1 \\ - 2 + 2 x & x > 1 \end{cases}$

Write the Absolute Value as Piecewise f(x)=|2-2x|

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