How to remove absolute value from inequality

Question

So I know that $|x| \le 2 \iff x\le 2 \text{ and } x \ge -2$. But, I was wondering would the same rules apply for $|x+2| + |y| \le 5$. Would it be correct to say that this inequality is equivalent to $x + y \le 3 \text{ and }x + y \ge -7$?

Answer 1

For a problem involving two variables such as $$ |x+2| + |y| \le 5$$ you need to look at 4 regions in the $xy-plane$

1) $y\le 0$ and $x+2 \le 0$. In this region your inequality is $-(x+2)-y \le 5$

2) $y\le 0$and $x+2 \ge 0$. In this region your inequality is $(x+2)-y \le 5$

3) $y\ge 0$ and $x+2 \ge 0$. In this region your inequality is $(x+2)+y \le 5$

4) $y\ge 0$ and $x+2 \le 0$. In this region your inequality is $-(x+2)+y \le 5$

Answer 2

Well, you decompose your inequality $|x+2| + |y| \le 5$ into intervals:

• $y<0 \implies |y| = -y$ and otherwise we have $|y|=y$
• $x<-2 \implies |x+2| = -(x+2)$ and otherwise $|x+2| = x+2$.

This gives you $2 \times 2 = 4$ interval ranges, each of which implies a different inequality.