# How do you write f(x)= |7/6x+4/3| as a piecewise function?

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Jan 24, 2018

See below.

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The definition of absolute value.

$| a | = a \textcolor{w h i t e}{8888888}$ If and only if $\textcolor{w h i t e}{8888888} a \ge 0$

$| a | = - a \textcolor{w h i t e}{8888}$ If and only if $\textcolor{w h i t e}{8888888} a < 0$

First we recognise that if $\frac{7}{6} x + \frac{4}{3} < 0$ , we have $- \left(\frac{7}{6} x + \frac{4}{3}\right)$

And if $\frac{7}{6} x + \frac{4}{3} \ge 0 \textcolor{w h i t e}{88}$ we have $\textcolor{w h i t e}{888} \left(\frac{7}{6} x + \frac{4}{3}\right)$

$\frac{7}{6} x + \frac{4}{3} < 0 \implies x < - \frac{8}{7}$

$\frac{7}{6} x + \frac{4}{3} \ge 0 \implies x < - \frac{8}{7}$

So piecewise we have:

$f \left(x\right) = \left[\begin{matrix}- \frac{7}{6} x - \frac{4}{3} & \to x < - \frac{8}{7} \\ \null & \null \\ \frac{7}{6} x + \frac{4}{3} & \to x \ge - \frac{8}{7}\end{matrix}\right]$

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