Expert: Ahmed Salami - 10/1/2009

Question
what is |2x+1|-|x-3|=10

Answer
Hi Avin,
|2x+1| - |x-3| = 10
The easy way to do this is to consider the points where the arguments (in the absolute sign) change sign i.e equal to zero.
For 2x + 1 = 0
x = -1/2
For x - 3 = 0
x = 3
We now split the entire number line into three intervals and consider what happens in each. The three intervals would be (-infinity, -1/2), (-1/2, 3) and (3, infinity).
In the first interval i.e x < -1/2
|2x+1| = -(2x+1)
|x-3| = -(x-3)
Back to the equation,
|2x+1| - |x-3| = 10
-(2x+1) - -(x-3) = 10
-(2x+1) + (x-3) = 10
-2x - 1 + x - 3 = 10
-x - 4 = 10
-x = 14
x = -14
This is considered a valid solution since it lies in the supposed interval (-infinity, -1/2).
In the second interval i.e -1/2 < x < 3
|2x+1| = 2x+1
|x-3| = -(x-3)
Back to the equation,
|2x+1| - |x-3| = 10
(2x+1) - -(x-3) = 10
(2x+1) + (x-3) = 10
2x + 1 + x - 3 = 10
3x - 2 = 10
3x = 12
x = 4
This is not a valid solution because it doesn't lie in the supposed interval (-1/2, 3)
In the third interval i.e x > 3
|2x+1| = 2x+1
|x-3| = x-3
Back to the equation,
|2x+1| - |x-3| = 10
(2x+1) - (x-3) = 10
2x + 1 - x + 3 = 10
x + 4 = 10
x = 6
Which is another valid solution because it lies in the supposed interval (3, infinity).

The complete solution is therefore x = -14 or 6
You can always get back to me.

Regards