Answer:
x ∈ [0, 2] ∪ [8, 10]
Explanation:
You want the solution to the inequality ll x-5 l -4 l ≤ 1.
Layer 1
We can write the original inequality as ...
-1 ≤ | x-5 | -4 ≤ 1
Adding 4 transforms this to two more inequalities:
3 ≤ | x-5 | ≤ 5
Layer 2a
The one on the right can be further expanded to ...
-5 ≤ x -5 ≤ 5
0 ≤ x ≤ 10 . . . . . . solution values will lie between 0 and 10, inclusive
Layer 2b
The inequality from Layer 1 on the left transforms to two disjoint inequalities:
x -5 ≥ 3 ⇒ x ≥ 8 . . . . . . . for x ≥ 5
x -5 ≤ -3 ⇒ x ≤ 2 . . . . . . for x ≤ 5
Solution
The intervals from Layer 2b partially overlap the interval from Layer 2a, so the final solution is ...
0 ≤ x ≤ 2 ∪ 8 ≤ x ≤ 10
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Additional comment
As in the attached graph, it is often useful to write the inequality as a comparison to zero. Here, we can do this by subtracting 1 from both sides of the original.