Exercise 22 Page 542

We are asked to find the solution set for all possible whole-number values of x in the given inequality. |x-5| ≤ 8 To do this, we will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than or equal to 8 away from the midpoint in the positive direction and any number less than or equal to 8 away from the midpoint in the negative direction. Absolute Value Inequality:& |x-5| ≤ 8 Compound Inequality:& - 8 ≤ x-5 ≤ 8We can split this compound inequality into two cases, one where x-5 is greater than or equal to - 8 and one where x-5 is less than or equal to 8. x-5≥- 8 and x-5 ≤ 8 Let's isolate x in both of these cases before graphing the solution set.

Case 1

This inequality tells us that all values less than or equal to 13 will satisfy the inequality.

Case 2

This inequality tells us that all values greater than or equal to - 3 will satisfy the inequality.

Solution Set

The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality. First Solution Set:& x ≤ 13 Second Solution Set:& - 3 ≤ x Intersecting Solution Set:& - 3 ≤ x ≤ 13

Graph

The graph of this inequality includes all values from - 3 to 13, inclusive. We show this by using close circles on the endpoints.

As we can see on the graph, the inequality has 17 whole solutions.