The largest real value of a for which the equation |x+a| + |x-1| = 2 has an infinite number of solutions for x is

Question

UPSC Quizroom · Accepted Answer

Consider different regions of x:

Case 1: x ≥ 1
Then |x + a| + |x - 1| = (x + a) + (x - 1) = 2x + a - 1
This is linear in x, so it can’t have infinitely many solutions.

Case 2: x ≤ -a
Then |x + a| + |x - 1| = -(x + a) + (1 - x) = -2x - a + 1
Also linear — can’t have infinitely many solutions.

Case 3: -a < x < 1
Then |x + a| = x + a and |x - 1| = 1 - x
⇒ |x + a| + |x - 1| = (x + a) + (1 - x) = a + 1

This expression is constant (does not depend on x).
For infinitely many solutions, we must have
a + 1 = 2
⇒ a = 1

Hence, the largest real value of a for which the equation has infinitely many solutions is 1.

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