Solving an absolute value inequality without using a certain property of the absolute value.

Question

Can I solve this inequality: $$|2x + 3| \leq 4$$ without using the property that $|x| \leq a$ is equivalent to $-a \leq x \leq a$ ?

Thanks!!

Answer 1

Hint: Look at

$2x+3>0 \implies 2x+3\leq 4$

and

$2x+3\leq 0 \implies -(2x+3)\leq 4$

Answer 2

You can do it by splitting the cases:

• case 1: $2x+3>0$: In this case, $|2x+3|=2x+3$
• case 2: $2x+3<0$: In this case, $|2x+3|=-(2x+3)$

Answer 3

$|x-a|$ it's a distance between $x$ and $a$ on the $x$-axis.

We need to solve that $$|2x+3|\leq4$$ or $$\left|x-\left(-\frac{3}{2}\right)\right|\leq2,$$ which gives $$-3.5\leq x\leq0.5.$$ Draw it!