If 2x - 3(2x - 2) > x - 1 < 2 + 2x; then x can take which of the following…

Concept: A chained (compound) inequality of the form A > B < C is shorthand for two simultaneous conditions: A > B and B < C. The value of the variable must satisfy both conditions at the same time, so the solution set is the intersection of the two individual solution sets, not the union.

Applying this to the given inequality:

1. Split the compound inequality into two separate parts: 2x − 3(2x − 2) > x − 1 and x − 1 < 2 + 2x.

2. Simplify the first part: 2x − 6x + 6 > x − 1, which gives −4x + 6 > x − 1, then 7 > 5x, so x < 7/5.

3. Simplify the second part: x − 1 < 2 + 2x, which gives −1 − 2 < 2x − x, so −3 < x, i.e. x > −3.

4. Intersecting both conditions, x must satisfy −3 < x < 7/5 at the same time.

Cross-check: Among the given options, only −2 lies inside the interval (−3, 7/5). Substituting x = −2 back into the original parts gives 14 > −3 and −3 < −2, so both conditions hold, confirming this value.