For what values of x is the following inequation valid : 2x^2 + x - 3

We are given the following inequation to solve 2x^2 + x -
3 < 0

2x^2 + x - 3 <
0

=> 2x^2 + 3x -2x -3 <
0

=> x ( 2x +3) -1 (2x + 3)
<0

=> (x - 1)(2x + 3)
<0

Now as ( x - 1)(2x + 3) is less than 0, either
 (x - 1) < 0 and (2x + 3) >0 or (x - 1) > 0 and (2x + 3) <
0.

For the first case:

(x - 1)
< 0 and (2x + 3) >0

=> x < 1
and x > -3/2

Therefore x takes values less than 1
and greater than -3/2 , or x lies in (-3/2 , 1)

For the
second case:

(x - 1) > 0 and (2x + 3) <
0

=> x > 1 and x <
-3/2

This implies x has to be simultaneously greater than 1
and less than -3/2, which is not possible.

So we cannot
obtain any values for x here.

Therefore the
possible values for x are ( -3/2 , 1)