2210.41 – Distance on the Number Line

The locus of points PP on the number line such that the distance between PP and the point 22 is between 11 and 77 is

  1. x≤1x \leq 1 or x≥3x \geq 3

  2. 1≤x≤31 \leq x \leq 3

  3. −5≤x≤9-5 \leq x \leq 9

  4. −5≤x≤1,-5 \leq x \leq 1, or 3≤x≤93 \leq x \leq 9

  5. −6≤x≤1-6 \leq x \leq 1 or 3≤x≤103 \leq x \leq 10

Solution

The answer is (d). Find it by drawing it! The figure far below shows how.

Or use algebra. The distance between points xx and yy on a line is ∣x−y∣|x - y|, so the given condition on PP is that 1≤∣P−2∣≤71\leq |P - 2|\leq 7. Let's take the two parts of this separately and see what they mean.

In the first place, 1≤∣P−2∣1\leq |P - 2| means either 1≤P−21 \leq P - 2 or P−2≤−1.P - 2 \leq -1. In other words, either 3≤P 3\leq P or P≤1.P\leq 1. On the other hand, ∣P−2∣≤7|P - 2|\leq 7 means that −7≤P−2≦7, -7 \leq P - 2\leqq 7, and this means −5≤P≤9. -5 \leq P\leq 9. Putting these together (which probably means drawing the picture), we see that either 3≤P≤93\leq P\leq 9 or −5≤P≤−3.-5\leq P\leq -3. This is (d).