How do you write $f (x) = ∣ ∣ ∣ \frac{7}{6} x + \frac{4}{3} ∣ ∣ ∣$ $f(x)= |7/6x+4/3|$ as a piecewise function?

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Answer 1 · 2018-01-24T18:22:38

See below.

The definition of absolute value.

$| a | = a 8888888$ $|a|=acolor(white)(8888888)$ If and only if $8888888 a \geq 0$ $color(white)(8888888)a>=0$

$| a | = - a 8888$ $|a|=-acolor(white)(8888)$ If and only if $8888888 a < 0$ $color(white)(8888888)a<0$

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

And if $\frac{7}{6} x + \frac{4}{3} \geq 088$ $7/6x+4/3>=0color(white)(88)$ we have $888 (\frac{7}{6} x + \frac{4}{3})$ $color(white)(888)(7/6x+4/3)$

$\frac{7}{6} x + \frac{4}{3} < 0 \Rightarrow x < - \frac{8}{7}$ $7/6x+4/3<0=>x<-8/7$

$\frac{7}{6} x + \frac{4}{3} \geq 0 \Rightarrow x < - \frac{8}{7}$ $7/6x+4/3>=0=>x<-8/7$

So piecewise we have:

$f(x)=[(-7/6x-4/3 , ->x<-8/7),(,),(7/6x+4/3,->x>=-8/7)]$

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How do you write f(x)= |7/6x+4/3|f(x)=∣∣∣76x+43∣∣∣f(x)= |7/6x+4/3| as a piecewise function?