Mathematics

The key is recognizing that absolute value functions create "corners" where their behavior changes, and the minimum of a sum of distances occurs at one of these critical points. Think of $|x-3|+|x+1|$ as the total distance from $x$ to the points 3 and −1 on a number line. When $x$ lies between these two points, the sum equals their separation; outside this interval, the sum grows linearly. By checking the behavior in each region, you'll find the minimum value and see how the function increases without bound.

The key is recognizing that absolute value functions create "corners" where their behavior changes, and the minimum of a sum of distances occurs at one of these critical points. Think of $|x-3|+|x+1|$ as the total distance from $x$ to the points 3 and −1 on a number line. When $x$ lies between these two points, the sum equals their separation; outside this interval, the sum grows linearly. By checking the behavior in each region, you'll find the minimum value and see how the function increases without bound.