Expert: Ahmed Salami - 10/10/2010

Question
find the range of values of x for which the common ratio r of a convergent geometric series is (2x-3)/(x+4)
THANK YOU VERY MUCH:-)

Answer
Hi Kristina,
The common ratio r of a convergent geometric series is such that |r| < 1. Therefore,
|(2x-3)/(x+4)| < 1
|(2x-3)| < |(x+4)|
Now, 2x-3 changes sign at
2x - 3 = 0 i.e x = 3/2
x+4 changes sign at
x+4 = 0 i.e x = -4
So we divide the number line into three intervals according to the points.
In the interval x < -4;
|2x-3| = -(2x-3)
|x+4| = -(x+4)
and then
-(2x-3) < -(x+4)
2x-3 > x+4
x > 7
but this solution doesnt lie in the interval and so we discard it.

In the interval -4 < x < 3/2;
|2x-3| = -(2x-3)
|x+4| = x+4
and then
-(2x-3) < x+4
2x-3 > -(x+4)
2x-3 > -x-4
3x > -1
x > -1/3
which is a valid solution since it lies in the considered interval.

In the interval x > 3/2;
|2x-3| = 2x-3
|x+4| = x+4
and then
2x-3 < x+4
x < 7
which is also valid.

Combining the two valid solutions, we get the complete solution of
-1/3 < x < 7

Regards