Question

Question #5984e

Answer 1

`(x+4)(2x-3) <= 0`

Explanation:

We are given the quadratic inequality

`2x^2 + 5x - 12 <= 0`

Split the middle term as shown below:

`2x^2 + 8x - 3x - 12 <= 0`

`2x(x+4) - 3(x+4) <= 0`

Hence, we can conclude that

`(2x - 3) (x + 4) <=0`

is what we get after factorizing the given quadratic inequality.

Answer 2

`x<=3/2` and `x<=-4`

Explanation:

.

`2x^2+5x-12<=0`

Let's add and subtract `3x` to the equation:

`2x^2+3x-3x+5x-12<=0`

`2x^2-3x+8x-12<=0`

`x(2x-3)+4(2x-3)<=0`

`(2x-3)(x+4)<=0`

`2x-3<=0` This gives us `x<=3/2`

`x+4<=0` This gives us `x<=-4`

Solution includes all of `(-4,3/2)`

Answer 3

Please see below.

Explanation:

`2x^2+5x-12 = 0` when

`(2x-3)(x+4)=0` which happens at

`x=3/2` and at `x=-4`

These numbers partition the number line into the intervals:

`(-oo,-4)` `" "` `(-4,3/2)` `" "` `(3/2,oo)`

Pick a number in each interval and test the inequality.

I'll use `-10`, `0` and `10`

At `x = -10`, we have
`(2x-3)(x+4)=(2(-10)-3)((-10)+4)` which is a negative times a negative, so it is positive and the inequality is false.

At `x = 0`, we have
`(2x-3)(x+4)=(2(0)-3)((0)+4)` which is a negative times a positive, so it is negative and the inequality is true.
The solutions include all of `(-4,3/2)`.

At `x = 10`, we have
`(2x-3)(x+4)=(2(10)-3)((10)+4)` which is a positive times a positive, so it is positive and the inequality is false.

The solution set is

`(-4,3/2)`.