# How do you write y= abs(3x+6) as a piecwise function?

Aug 2, 2017

See a solution process below:

#### Explanation:

Step 1) First, solve the term within the absolute value function for $0$:

$3 x + 6 = 0$

$3 x + 6 - \textcolor{red}{6} = 0 - \textcolor{red}{6}$

$3 x + 0 = - 6$

$3 x = - 6$

$\frac{3 x}{\textcolor{red}{3}} = - \frac{6}{\textcolor{red}{3}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} x}{\cancel{\textcolor{red}{3}}} = - 2$

$x = - 2$

Step 2) Multiply the term within the absolute value function by $- 1$ and write a "less than" inequality with the result of Step 1:

$- 1 \left(3 x + 6\right) \implies - 3 x - 6$

$y = - 3 x - 6 \text{ for } x < - 2$

Step 3) Take the term within the absolute value function and write a "greater than or equal to" inequality with the result of Step 1:

$y = 3 x + 6 \text{ for } x \ge - 2$

Step 4) Combine Step 2 & Step 3 to form the piecewise function:

$y = \left\{- 3 x - 6 \text{ for "x < -2; 3x + 6" for } x \ge - 2\right\}$