What are the integer solutions to the absolute value inequality |X - 2| ≤ 5?

To solve the absolute value inequality |X - 2| ≤ 5, we begin by interpreting what this inequality means. An absolute value inequality of the form |A| ≤ B indicates that the expression A is bounded between -B and B. In this case, A is X - 2, and B is 5.

The inequality can be rewritten without the absolute value as two simultaneous inequalities:

-5 ≤ X - 2 ≤ 5.

To solve this, we will break it down into two parts.

First, consider the left part of the inequality:

X - 2 ≥ -5.

Adding 2 to both sides gives:

X ≥ -3.

Next, consider the right part of the inequality:

X - 2 ≤ 5.

Adding 2 to both sides gives:

X ≤ 7.

Now we can combine these two results:

-3 ≤ X ≤ 7.

This indicates that X can take on any integer value from -3 to 7, inclusive. The integer solutions within this specified range are -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, and 7.

This set of