Expert: Ahmed Salami - 8/22/2010

Question
Find THe solution set of the equation |2x+3|-|x-1|= 6

Answer
Hi Sanjana,
2x + 3 changes sign at 2x + 3 = 0
i.e x = -3/2
x - 1 changes sign at x - 1 = 0
i.e x = 1
These two points divide the number line into three intervals;
x < -3/2, -3/2 < x < 1, x > 1
In the first interval,
|2x + 3| = -(2x + 3)
|x - 1| = -(x - 1)
The equation becomes
-(2x + 3) - -(x - 1) = 6
-(2x + 3) + (x - 1) = 6
-2x - 3 + x - 1 = 6
-x - 4 = 6
-x = 10
x = -10
which is consistent since the x = -10  falls in the selected interval.

In the second interval,
|2x + 3| = (2x + 3)
|x - 1| = -(x - 1)
The equation now becomes
(2x + 3) - -(x - 1) = 6
(2x + 3) + (x - 1) = 6
2x + 3 + x - 1 = 6
3x + 2 = 6
3x = 4
x = 4/3
which falls outside the selected interval and so is not a solution.

In the third interval,
|2x + 3| = (2x + 3)
|x - 1| = (x - 1)
The equation becomes
(2x + 3) - (x - 1) = 6
(2x + 3) - (x - 1) = 6
2x + 3 - x + 1 = 6
x + 4 = 6
x = 2
which is also consistent since as it falls in the selected interval.

Regards