Question

Use the Distance Formula and the x-axis of the coordinate plane. Show why the distance between two points on a number line (the x-axis) is | a – b |, where a and b are the x-coordinates of the points.

Answer 1

Let \((a,0)\) and \((b,0)\) be two distinct points on the \(x\)-axis, i.e. the real number line. The distance between these points is
\[d=\sqrt{(a-b)^2+(0-0)^2}=\sqrt{(a-b)^2}=|a-b|\]
We take the absolute value because it's possible that \(b>a\), which would make \(a-b<0\). However, the square root is undefined for negative arguments.