CAT 2022 Slot 1 — QA Question 15

Interpret geometrically. $|x-1|$ is the distance from $x$ to $1$, and $|x+a|=|x-(-a)|$ is the distance from $x$ to $-a$:

$$|x+a|+|x-1|=\text{dist}(x,-a)+\text{dist}(x,1)$$

When is the sum constant? For any $x$ lying between the two points, the two distances add up to exactly the distance between the points. For this constant to equal $2$:

$$|{-a}-1|=2\ \Rightarrow\ |a+1|=2$$

Solve and pick the largest.

$$a+1=\pm 2\ \Rightarrow\ a=1\ \text{ or }\ a=-3$$

The largest real value is $a=1$ (points $-1$ and $1$, distance $2$).