Write the Absolute Value as Piecewise |3x-9|

$| 3 x - 9 |$

Step 1

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

$3 x - 9 \geq 0$

Step 2

Solve the inequality.

Add $9$ to both sides of the inequality.

$3 x \geq 9$

Divide each term in $3 x \geq 9$ by $3$ and simplify.

Divide each term in $3 x \geq 9$ by $3$ .

$\frac{3 x}{3} \geq \frac{9}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{9}{3}$

Divide $x$ by $1$ .

$x \geq \frac{9}{3}$

Simplify the right side.

Divide $9$ by $3$ .

$x \geq 3$

Step 3

In the piece where $3 x - 9$ is non-negative, remove the absolute value.

$3 x - 9$

Step 4

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

$3 x - 9 < 0$

Step 5

Solve the inequality.

Add $9$ to both sides of the inequality.

$3 x < 9$

Divide each term in $3 x < 9$ by $3$ and simplify.

Divide each term in $3 x < 9$ by $3$ .

$\frac{3 x}{3} < \frac{9}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{9}{3}$

Divide $x$ by $1$ .

$x < \frac{9}{3}$

Simplify the right side.

Divide $9$ by $3$ .

$x < 3$

Step 6

In the piece where $3 x - 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 x - 9)$

Step 7

Write as a piecewise.

${\begin{cases} 3 x - 9 & x \geq 3 \\ - (3 x - 9) & x < 3 \end{cases}$

Step 8

Simplify $- (3 x - 9)$ .

Apply the distributive property.

${\begin{cases} 3 x - 9 & x \geq 3 \\ - (3 x) - - 9 & x < 3 \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x - 9 & x \geq 3 \\ - 3 x - - 9 & x < 3 \end{cases}$

Multiply $- 1$ by $- 9$ .

${\begin{cases} 3 x - 9 & x \geq 3 \\ - 3 x + 9 & x < 3 \end{cases}$

Write the Absolute Value as Piecewise |3x-9|

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