Write the Absolute Value as Piecewise |-6-7x|

$| - 6 - 7 x |$

Step 1

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

$- 6 - 7 x \geq 0$

Step 2

Solve the inequality.

Add $6$ to both sides of the inequality.

$- 7 x \geq 6$

Divide each term in $- 7 x \geq 6$ by $- 7$ and simplify.

Divide each term in $- 7 x \geq 6$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{6}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{6}{- 7}$

Divide $x$ by $1$ .

$x \leq \frac{6}{- 7}$

Simplify the right side.

Move the negative in front of the fraction.

$x \leq - \frac{6}{7}$

Step 3

In the piece where $- 6 - 7 x$ is non-negative, remove the absolute value.

$- 6 - 7 x$

Step 4

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

$- 6 - 7 x < 0$

Step 5

Solve the inequality.

Add $6$ to both sides of the inequality.

$- 7 x < 6$

Divide each term in $- 7 x < 6$ by $- 7$ and simplify.

Divide each term in $- 7 x < 6$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} > \frac{6}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} > \frac{6}{- 7}$

Divide $x$ by $1$ .

$x > \frac{6}{- 7}$

Simplify the right side.

Move the negative in front of the fraction.

$x > - \frac{6}{7}$

Step 6

In the piece where $- 6 - 7 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 6 - 7 x)$

Step 7

Write as a piecewise.

${\begin{cases} - 6 - 7 x & x \leq - \frac{6}{7} \\ - (- 6 - 7 x) & x > - \frac{6}{7} \end{cases}$

Step 8

Simplify $- (- 6 - 7 x)$ .

Apply the distributive property.

${\begin{cases} - 6 - 7 x & x \leq - \frac{6}{7} \\ - - 6 - (- 7 x) & x > - \frac{6}{7} \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} - 6 - 7 x & x \leq - \frac{6}{7} \\ 6 - (- 7 x) & x > - \frac{6}{7} \end{cases}$

Multiply $- 7$ by $- 1$ .

${\begin{cases} - 6 - 7 x & x \leq - \frac{6}{7} \\ 6 + 7 x & x > - \frac{6}{7} \end{cases}$

Write the Absolute Value as Piecewise |-6-7x|

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