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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to attach the equation to the given inequality, such that:

`(x+3)(2x-3) = 0 => {(x + 3 = 0),(2x - 3 = 0):} => {(x = -3),(x = 3/2):}`

You need to consider a value for `x in (-3,3/2)` , such that:

`x = 0 => (0+3)(2*0-3) = -9 < 0`

You need to consider a value for `x < -3` such that:

`x = -4 => (-4+3)(-8-3) = 11 > 0`

You need to consider a value for `x > 3/2` such that:

`x = 2 => (2+3)(4-3) = 5 > 0`

Hence, the expression `(x+3)(2x-3)` is negative if `-3 < x < 3/2.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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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