Question 24

Explanation

Define each absolute value

It is seen that the boundary values of $x$ from both absolute values are $x = -\frac{1}{2}$ and $x = 0$. This means the boundaries will form three regions, namely $\text{Region I } (x < -\frac{1}{2})$, $\text{Region II } (-\frac{1}{2} \leq x < 0)$, and $\text{Region III } (x \geq 0)$. Based on the regions and their definitions, the problem can be solved.

$\text{Region I:}$ $(x < -\frac{1}{2})$

The solution for Region I is $\{x < -\frac{1}{2}\} \cap \{x > -\frac{3}{2}\} = \{-\frac{3}{2} < x < -\frac{1}{2}\}$.

$\text{Region II:}$ $(-\frac{1}{2} \leq x < 0)$

All $x$ in Region II are true, so the solution is $\{-\frac{1}{2} \leq x < 0\}$.

$\text{Region III:}$ $(x \geq 0)$

The solution is $\{x \geq 0\} \cap \{x < 1\} = \{0 \leq x < 1\}$.

So the overall solution is the union