Express $|a+b|-|b|$ without absolute value signs I am having trouble understanding what cases I need to evaluate. So far I've checked $a = b = 0$ and that results in the expression being equal to $0$. I've checked $0 \le b < a$ which results in $a$. I have checked seven other cases which have all resulted in $a$ or $0$. I checked the answer in the back of the book and it has only four cases (with a couple more different results):
$a \ge -b$ and $b \ge 0$,
$a \le -b$ and $b \le 0$,
$a \ge -b$ and $b \le 0$, and 
$a \le -b$ and $b \ge 0$.
My question is how does one know which cases to check? I tested a bunch of cases and still missed other cases. It also appears that I needn't check so many cases when there apparently are four cases to check which cover all possibilities.
 A: Since there is a $|b|$ in your expression, you may check $b\geq0$ and $b<0$. Additionally, since there is a $|a+b|$ in your expression, you can check $a+b\geq0\,(\iff a\geq-b)$ and $a+b<0\,(\iff a<-b)$.
These in all give $4$ cases:
For $b\geq0$, we either have $a\geq-b$ or $a<-b$.
For $b<0$, we again either have $a\geq-b$ or $a<-b$.
A: Recall that the definition of $|x|$ is piecewise: 
$$|x| = \left\{
  \begin{array}{lr}
    -x & : x \leq 0\\
    x & : x \ge 0
  \end{array}
\right.
$$
We check each occurrence of $|x|$ in your equation and check each case as to whether the $x$ is negative or not. 
For $|b|$ this is $b\leq0$ and $b \geq 0$; for $|a+b|$ this is $a+b\leq 0 \iff a\leq-b$ and $a+b\geq 0 \iff a\geq -b$.
A: The trick is to know where the absolute value function changes course. $ |x| $ has two different values one for $x \ge 0$ and another for $ x \lt 0$. So generally to find which cases you need to distinguish what you need to do is to find the points where the absolute value function changes course. Hence for an example equate $|a + b|$ and $|b|$ to $0$. Then the cases to be considered are 
$a + b \ge  0, \;  a + b \lt  0, \; b \ge 0$ and $b \lt 0$
But from here onwards you need to pair up the cases andthat is why you have $4$. 
