Sri,

Be careful: When a DS question asks you for the value of an expression, SUFFICIENT information means that you have enough to know the exact numerical value. So in general, proving that |x|+x = 2x is not enough to say that you have sufficient info to find |x|+x, unless you know exactly what number it is equal to.

For instance, if a DS asked "what is the value of |x|+x?" and a statement said "|x|+x = 2x", that statement would not be sufficient because we are unable to isolate a unique numerical value (if x=2 then |x|+x=4, but if x=1 then |x|+x=2)

Here is how I would approach the question:

If x is an integer, what is the value of x^2 + x ?

(1) x^2 = |x|

(2) x^2 + x > 0

Statement 1
For almost all integers, x^2 will be bigger than |x|. The only integers for which x^2=|x| are -1, 0, and 1 (visualize the absolute value "V" graph, and the x^2 parabola graph on a plot; if you know what the graphs look like you will realize right away that there are just 3 points of intersection). So this statement tells us that x = {-1, 0 or 1}. x=-1 would make x^2+x = 2, but x=0 or -1 would make x^2+x = 0. INSUFFICIENT

Statement 2
INSUFFICIENT since we could get lots of results for |x|+x depending on what we make x.

Together
Statement 1 limited us to only 2 possible values for x^2+x (2 or 0). Statement 2 tells us that x^2+x > 0, so together the statements guarantee that x^2+x = 2. SUFFICIENT

-Patrick

The solutions above are great.
I do not recommend using algebra to evaluate statement 1.
But if really insist, here's one approach:

x² = |x|
|x||x| = |x|
|x||x| - |x| = 0
|x| (|x| - 1) = 0.

|x| = 0 when x=0.
|x| - 1 = 0 when x=-1 or x=1.
Thus, valid solutions for statement 1 are x=-1, x=0, x=1.