List all integers satisfying the inequality in -2 < 2x-6 < 4

Analyze the question: We need all integers x for which 2x − 6 lies strictly between −2 and 4.

Concept: A compound inequality can be solved by doing the same operation on all three parts. The inequality signs are strict (< not ≤), so the end values are not included.

Solution:

−2 < 2x − 6 < 4

Add 6 to all parts: 4 < 2x < 10

Divide all parts by 2: 2 < x < 5

The integers strictly between 2 and 5 are 3 and 4.

Check:

• x = 3: 2(3) − 6 = 0, and −2 < 0 < 4 ✓
• x = 4: 2(4) − 6 = 2, and −2 < 2 < 4 ✓
• x = 2: 2(2) − 6 = −2, and −2 < −2 is false ✗
• x = 5: 2(5) − 6 = 4, and 4 < 4 is false ✗

Answer: D (3, 4)

Tip: With strict inequalities (< or >), always drop the boundary values. Testing the endpoints, as in the check above, quickly eliminates options like A, B and C.