For what x the inequality is true 2(x-1/2)(x+2)<0

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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).

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