Inequality Case by Case way I just wanted to know the rule of evaluating an in-equation.
How do we know  the  number cases we have to take into account ? 


*

*For example


if we have   $-\frac{x}{|4x+1|} ≤ \frac{x}{(x+2)}$ ?     
for what values of x will we have to look at ?
I know for sure  $x < -2$,
$-2 < x < -\frac14$ 
but I'm not sure about  for $x = 0$. ?
 A: You normally include the breaking points of absolute value expressions (here, $|4x+1|=0$ when $x=-1/4$) and denominators (here $x+2=0$ when $x=-2$).
Therefore, you will have 3 intervals of interest: $(-\infty,-2),(-2,-1/4),(-1/4,\infty)$ plus the boundary points $-2, -1/4$ which need to be tested as well.
EDIT Forgot to include break points of all other stand-alone terms, here $x$ breaks at $0$. So the final interval list is $(-\infty,-2),(-2,-1/4),(-1/4,0),(0,+\infty)$
A: We need to look at the denominator (especially their signs). In order to solve your inequality, you don't need to look at the numerator.
By the way, if $x=0$, then the equality holds.
A: First you have to check for 

$x \ge -\frac14 $, so $|4x+1| = 4x+1$

Now this removes the mod and you can solve the inequality by bringing all the terms to one side and expressing them as linear factors. Now take intersection of the resultant solution with $x \ge -\frac14$.This will give you a part of your final solution.
Now do the same for $x \lt -\frac14$ and proceed like above. 
Now take union of both the solutions. This will give you your final answer.
It does not matter whether you take 
$x \ge \frac14$ and $x \lt \frac14$ 
                            OR

$x \gt \frac14$ and $x \le \frac14$
