Final answer:
The solution to the inequality -2 < [x] ≤ 7 is any real number x that is greater than -1 and less than or equal to 7, as these values satisfy the condition set by the greatest integer function.
Step-by-step explanation:
The student is asking to find the set of real numbers x that satisfy the inequality -2 < [x] ≤ 7, where [x] denotes the greatest integer function or floor function of x. This function gives the largest integer less than or equal to a given number. Since the inequality states that [x] is greater than -2, but less than or equal to 7, we look at the integers that this range includes. These integers are -1, 0, 1, 2, 3, 4, 5, 6, and 7. The corresponding intervals for x will be all the numbers greater than -1 but less than or equal to 7.
Therefore, the solution to the inequality is x > -1 and x ≤ 7. This means any number greater than -1 and up to and including 7 will satisfy the initial inequality.