No need to work out so much. You can tell the answer just by looking at the question. For x-|y| to be maximum |y| should be minimum and |x| should be maximum. minimum value of y=0. Put y=0 in second equation. It satisfies, so y=0. Now x can be either -4 or 7. maximum value =7.
So 7-0= 7 Ans.
Your doing everything right except the last condition :
-4y <= 14
=> y >= - 7/2
You're missing that "-" sign here. Thus, maximum value will be when x = 7 and y = 0
x - |y = 7 - 0 = 7 
[QUOTE=Omkarp.
y| >= 0. So, minimum value y will take is 0. It also satisfies the given inequality.
Now, just find the max value of x. 9+5 = 14. And 2*7 = 14. So, x<=7.
And so x| - |y <=7.
The problem with your approach is that you are trying to minimize y, not |y.
Hence, you are getting a different answer.
Hope it helps.

1st equation:
-9 <= 2x-5 <= 9
-9+5 <= 2x <= 14
-4 <= 2x <= 14
-2 <= x <= 7
So, x lies between -2 to 7
2nd equation
-21 <= 4y - 7 <= 21
-21 +7 <= 4y <= 21 +7
-14 <= 4y <= 28
-7/2 <= y <= 4
So, y lies between -7/2 to 4 ...
Hence, y can take a value of 0 also ...
So, max value of x - |y| will be x(max) - 0
i.e. 7-0 = 7 .
In your method you are making a mistake in the 4th equation.
I hope am correct and clear.
I now realised why i always keep complaining about my speed ... I easily wasted 2-3 mins on this question whereas puys here solved(seems) by simply looking at the question.
I will try using the pencil less in the next aimcat. Hope it brings a change in my speed and increases my number of attempts.
Any tips on how to increase the number of attempts ...
I attempted 16 with 15 c and 1 wrong.
I really need to increase my attempts ...
A minute's success pays the failure of years ...