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

$(2x + 3) (x + 1) = 0 \implies 2x+3 = 0\text{ or }x+1 = 0 \\\\ \implies x = -\dfrac32 \text{ or } x = -1$

So the solution to the inequality is the set

$\left\{x \in \Bbb R \mid x\neq-\dfrac32 \text{ and } x\neq-1\right\}$