Question

Solve the inequality (2x-1)(x+2)<0

Accepted Answer

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

since the value is negative then we have two cases:

case (1):

2x-1 <0 AND x+ 2 > 0

x < 1/2 AND x >-2

==> -2 < x < 1/2

==> x belongs to the interval (-2, 1/2)

Case 2:

2x-1 > 0 AND x+2 < 0

x > 1/2 AND x < -2

==> x = (1/2, inf) + (-inf, -2) = empty set.

Then the solution is :

x belongs to ( -2, 1/2)

Answer

(2x-1)(x+2)<0.
To solve the  inequality.
Solution:
First we consider the roots of the equation; (x+2)(2x-1) = 0.
x = -2 and x= 1/2/
The product on the left is less than zero if only one of the factors  is negative and the other is positive .
Product is positive if both of the factors are positive or both of the factors are negative.
So the product is negative  only if  the value of x is between the roots -2 and 1/2. Or x belongs to the interval (-2 , 1/2)

Answer

The product of two terms is negative if either of the terms is negative and the other is positive. For (2x-1)(x+2)<0 2x-1 < 0 and x+2 >0 => 2x < 1 and x > -2 => x < 1/2 and x > -2 So x lies between -2 and 1/2 2x - 1> 0 and x+2 <0 => 2x > 1 and x < -2 => x > 1/2 and x< 2 This is not possible. Therefore the solution is x lies between -2 and 1/2.

Answer

We'll conclude that a product is negative if the factors are of opposite sign. There are 2 caes of study: 1) (2x-1) < 0 and (x+2) > 0 We'll solve the firts inequality. For this reason, we'll isolate 2x to the left side. 2x < 1 We'll divide by 2: 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 systemof inequalities: 2) (2x-1) > 0 and (x+2) < 0 2x-1 > 0 We'll add 1 both sides: 2x > 1 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).

Math

Solve the inequality (2x-1)(x+2)<0

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