Verify if the inequality is true. Discuss! 2x^2 + 5x

First, we have to subtract 12, both sides of the equation
(it's easier to verify if an expression is negative or
positive).

2x^2 + 5x - 12 <
0

Now, it would be much more easier to factorize the
expression. In order to do so, we'll find out the roots of the equation 2x^2 + 5x - 12 =
0

x1 = [-5+sqrt(25+96)]/4

x1 =
(-5+11)/4

x1 = 6/4

x1 =
3/2

x2 = (-5-11)/4

x2 =
-16/4

x2 = -4

Now, we'll
factorize the expression 2x^2 + 5x - 12:

2x^2 + 5x - 12 =
2(x+4)(x-3/2)

2x^2 + 5x - 12 =
(x+4)(2x-3)

Now, we'll discuss when
(x+4)(2x-3)<0.

For a product to be negative, the
factors of the product have to have opposite signs. From here, it results 2
cases:

First
case:

x+4>0 and
2x-3<0

x>-4 and
x<3/2

x belongs to the interval
(-4,3/2).

Second
case:

x+4<0 and
2x-3>0

x<-4 and
x>3/2

It is obvious that there are no real numbers
to satisfy this condition,
simultaneously.

So, the inequality is
verified for x belongs to the interval (-4,3/2).