Question

Help please!

| 2x^2 + 5x + 3 | > 0
solve the inequality

Answer 1

Factorize the quadratic.

`2x^2 + 5x + 3 = (2x + 3) (x + 1)`

We have
`|ab| = |a||b|`, so

`|2x^2 + 5x + 3| = |2x + 3| |x + 1| > 0`

Now, both
`|2x+3|\ge0` and
`|x+1|\ge0` (since the absolute value of any number cannot be negative), so we just need to worry about when the left side is exactly zero. This happens for

`(2x + 3) (x + 1) = 0 \implies 2x+3 = 0\text{ or }x+1 = 0 \\\\ \implies x = -\frac32 \text{ or } x = -1`

So the solution to the inequality is the set

`\left\{x \in \Bbb R \mid x\\eq-\frac32 \text{ and } x\\eq-1\right\}`