16Hit1600
AlgebraHardLinear inequalities

Algebra practice question

For how many integer values of x is −2 < (x − 5)/3 ≤ 1 true?

Answer: 9

Multiply all parts by 3: −6 < x − 5 ≤ 3. Add 5: −1 < x ≤ 8. The integers strictly greater than −1 and at most 8 are 0,1,2,3,4,5,6,7,8 — that's 9 values. Watch the strict vs. non-strict signs on each end.

Common wrong answers

If you answered 3
You added 5 without first multiplying all three parts by 3, getting 3 < x ≤ 6. Multiply by 3 first: −6 < x − 5 ≤ 3.
If you answered 8
You dropped x = 8, but the right end is non-strict (x ≤ 8), so 8 counts; the answer is 9.
If you answered 10
You counted −1 as a solution, but the left inequality is strict (−1 < x), so x = 0 is the smallest integer — 9 values, not 10.

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