Solve the inequality  (x+3)(2x-3)<0

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

(x+3)(2x-3) < 0

Since the result of the product is negative, then only one of the terms should be negative:

Then:

(x+3)<0     and  (2x-3) > 0

==> x < -3    and    x > 3/2    (impossible)

OR:

(x+3) > 0    and  (2x-3) < 0

==> x > -3    and  x < 3/2

==> -3 < x < 3/2

then the solution is:

x belongs to the interval ( -3, 3/2) 

william1941's profile pic

william1941 | College Teacher | (Level 3) Valedictorian

Posted on

We have the inequality (x+3) (2x-3) <0.

Now (x+3) (2x-3) <0 when either of (x+3) and (2x-3) is less than 0 and the other is greater than 0.

First let's take (x+3)> 0 and (2x-3) <0

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

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

This gives x lies in the range where x > -3 and x< -3/2.

Next let's take x+3 <0 and 2x-3 > 0

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

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

This does not give valid values.

Therefore for x:  -3 < x < -3/2

 

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

(x+3)(x-3) < 0

The left is a product of 2 linear epression. The product is  less than zero. Or the product should be negative.

Therefore the values of  both factors cannot be both positive together or both negative together in which case the product becomes positive.

So the possibility is x + 3 > 0 and x-3 < 0.

So If the value of lies between -3 and +3 , then x+3 < 0 and x -3 < 0

Therefore in order that (x+3)(x-3) < 0, the values of should bel within the interval of (-3 ,3) of  the number line.

 

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

There are 2 cases of study:

1)  (2x-3) < 0

and

      (x+3) > 0

We'll solve the first inequality. For this reason, we'll isolate 2x to the left side.

2x < 3

We'll divide by 2:

x < 3/2

We'll solve  the 2nd inequality:

    (x+3) > 0

We'll subtract 3 both sides:

x > -3

The common solution of the first system of inequalities is the interval (-3 , 3/2).

We'll solve the second case  for the following system of inequalities:

2)  (2x-3) > 0

and

      (x+3) < 0

2x-3 > 0

We'll add 3 both sides:

2x > 3

x > 3/2

     (x+3) < 0

x < -3

Since there is not a common interval to satisfy 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 (-3 , 3/2).

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