For what x the inequality is true 2(x-1/2)(x+2)

We'll divide both sides by
2:

Since the value is positive, teh inequality still
holds:

(x -
1/2)(x+2)<0

We'll conclude that a product is
negative if the factors are of opposite sign.

There are 2
cases of study:

1)  (x - 1/2) <
0

and

      (x+2) >
0

We'll solve the first inequality. For this reason, we'll
isolate x to the left side.

x <
1/2

We'll solve  the 2nd
inequality:

    (x+2) >
0

We'll subtract 2 both
sides:

x > -2

The
common solution of the first system of inequalities is the interval (-2 ,
1/2).

We'll solve the second system of
inequalities:

2)  (x-1/2) >
0

and

      (x+2) <
0

x-1/2 > 0

x >
1/2

     (x+2) < 0

x
< -2

Since we don't have a common interval to satisy
both inequalities, we don't have a solution for the 2nd
case.

So, the complete solution is the
solution from the first system of inequalities, namely the interval (-2 ,
1/2).