If x ∈ ℤ and −10 < 2x ≤ 6, how many integer solutions are there?

Correct answer: 8

The governing method is to solve the compound inequality before counting the integers. Divide every part of −10 < 2x ≤ 6 by 2. Since 2 is positive, neither inequality sign changes, so −5 < x ≤ 3. The integer values satisfying this interval are −4, −3, −2, −1, 0, 1, 2 and 3. The lower endpoint −5 is excluded because the first sign is strict, whereas the upper endpoint 3 is included because the second sign permits equality. Counting the listed values gives 8 integer solutions, so option B is correct. A undercounts by one, while C and D include values that lie outside the permitted interval. The equivalent number-line picture has an open point at −5 and a closed point at 3.