Graph $y=|x+8|+|x-8|$

Question

Graph $y=|x+8|+|x-8|$

I tried to simply this with

$$y=(x+8)+(x-8) \implies y=2x,x>0\\ y=(-x+8)+(-x-8) \implies y=-2x,x<0$$

But this looks quite different from the original.

I look for a short and simple way.

I have studied maths up to $12$th grade.

Answer 1

When $x > 8$, then we have $y = (x+8) + (x - 8) = 2x$ since both terms are positive when $x>8$.

When $x \leq -8$, then we have $y = -(x+8) + -(x-8) = -2x$ since both terms are negative.

But when $-8 < x \leq 8$, then $y = x+8 + -(x-8) = 16$ since the first term is positive but the second is negative.

So collecting these together, we have

$$y = \begin{cases}2x & x > 8 \\ 16 & -8 < x \leq 8\\ -2x & x \leq -8 \end{cases}$$

Answer 2

HINT : You need to have three cases as

$$x\le -8,\ \ \ -8\lt x\le 8,\ \ \ 8\lt x.$$

And note that $$|x+8|=x+8$$ for $\color{red}{x+8}\ge 0$, i.e. $x\ge -8$.