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