Posted by jase on Friday, November 25, 2011 at 11:35pm.
|x - 2/3| <= 4
if x - 2/3 >= 0 then |x - 2/3| = x - 2/3
In that case,
x - 2/3 <= 4
x <= 14/3 and x >= 2/3 (remember x - 2/3 >= 0)
so, x is in [2/3 , 14/3]
if x - 2/3 >= 0 then |x - 2/3| = -(x - 2/3)
In that case,
2/3 - x <= 4
x >= -10/3
But x - 2/3 < 0 means x < 2/3
So,
x is in [-10/3 , 2/3)
So, finally, x is in [-10/3 , 2/3)U[2/3 , 14/3]
or, x is in [-10/3 , 14/3]
This makes sense. Think of the graph of |x|. It is a V shape. If |x| < k, then you want the part of the V below the line y=k.
In this case, the V is shifted 2/3 to the right, and we want the part of the V below the line y=4