<s>[INST] How to remove absolute value from inequality 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 $x + y \le 3 \text{ and }x + y \ge -7$?
 [/INST] 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 $-(x+2)+y \le 5$
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