Set 11: Linear Inequalities (Intermediate)

Choice A is correct. Choice A is the correct answer. Unwrap the nested absolute value. 1. Outer: $-2 \leq |2 x - 1| - 3 \leq 2$ 2. Add 3: $1\leq |2 x - 1| \leq 5$ 3. Split: Distance is between 1 and 5. - Case A: $1\leq 2 x - 1 \leq 5$ - Add 1: $2\leq 2 x \leq 6 \rightarrow 1 \leq x \leq 3$ - Case B: $-5 \leq 2 x - 1 \leq -1$ - Add 1: $-4 \leq 2 x \leq 0 \rightarrow -2 \leq x \leq 0$ 4. Combine: $[-2, 0] \cup [1, 3]$ Strategic Tip: $|u| \in [a, b]$ means $u \in [a, b]$ or $u \in [-b, -a]$. Choice B is incorrect because it includes the gap $(0, 1)$. Choice C is incorrect because it misses the negative interval. Choice D is incorrect because it includes large numbers.