How to subtract an absolute value from both sides of an inequality? ($y+|x|<3$)

Question

I have been trying to find an example online, but I can't.

If I had an equation like:

$$y + |x| < 3$$

and I wanted to graph it, I know that to get it to slope intercept form I need to isolate the $y$.

How do I subtract the $x$ though?

Does it become $-|x|$ on the right side or does it become $|-x|$?

Answer 1

$$y+|x| < 3$$ Subtracting $|x|$ from both sides, we have $$y+|x|-|x| < 3-|x|$$ Since $|x|-|x|=0$,

$$y < 3-|x|$$

That is for $x \geq 0$, plot $y <3-x$.

For $x<0$, plot $y < 3+x$.

@Daniel_W._Farlow's approach based on transformation is awesome.

Answer 2

You should think of "$|x|$" as a number, just like any other number: $3$, or $x$, or $(x + 5)$, or $10$. These are all numbers. You are allowed subtract any number from both sides of an inequality.

In this case, we have $$ y + |x| < 3 $$ and we have to subtract the same number from both sides, so we subtract "$|x|$" from both sides: $$ y + |x| - |x| < 3 - |x| $$ to arrive at $$ y < 3 - |x|. $$