To solve this inequality, we need to consider two cases: when the quantity inside the absolute value is positive and when it is negative.

Case 1: when 2x - 7 is non-negative (greater than or equal to 0)

2x - 7 ≥ 0
2x ≥ 7
x ≥ 7/2

Now, substitute this value of x back into the inequality to check if it satisfies the original inequality:

|2(7/2) - 7| ≤ 2
|7 - 7| ≤ 2
|0| ≤ 2
0 ≤ 2

This is true, so x ≥ 7/2 is a solution for this case.

Case 2: when 2x - 7 is negative

2x - 7 < 0
2x < 7
x < 7/2

Now, substitute this value of x back into the inequality to check if it satisfies the original inequality:

|2(7/2) - 7| ≤ 2
|7 - 7| ≤ 2
|0| ≤ 2
0 ≤ 2

Since this is always true, any x values less than 7/2 also satisfy the inequality.

Therefore, the solution to the inequality |2x-7| ≤ 2 is x ≤ 7/2 or x ≥ 7/2.