The locus of points *P* on the number line such that the distance between *P* and the point 2 is between 1 and 7 inclusive, is:

*P*≤ 1 or*P*≥ 3- 1 ≤
*P*≤ 3 - -5 ≤
*P*≤ 9 - -5 ≤
*P*≤ 1 or 3 ≤*P*≤ 9 - -6 ≤
*P*≤ 1 or 3 ≤*P*≤ 10

Draw it!

So (d).

Or translate it: 1 ≤ |*P* – 2| ≤ 7

We have two cases: *P* – 2 > 0 and *P* – 2 < 0
Case I: *P* – 2 > 0 and 1 ≤ *P* – 2 ≤ 7 → *P* > 2 and 3 ≤ *P* ≤ 9

Case II: *P* – 2 < 0 (so 2 - *P* > 0) and 1 ≤ 2 – *P* ≤ 7 → *P* < 2 and -1 ≤ -*P* ≤ 5

→ *P* < 2 and 1 ≥ *P* ≥ -5

So the answer is (d).