How many integers $x$ satisfy $-5<2x+1$ ≤ 7?

Subtract 1 throughout: $-6<2x$ ≤ 6. Dividing by 2 gives −3 < $x$ ≤ 3. The integers strictly greater than −3 and at most 3 are −2, −1, 0, 1, 2, 3 — six of them (note −3 itself is excluded but 3 is included).