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Previous Year Question (PYQs)

What is the number of non-zero integral solutions of the equation $\,|x| + |1 - x| = 6\,$?




Solution

Case 1: $x \ge 1 \Rightarrow |x|=x,\ |1-x|=x-1$ $\Rightarrow x + (x-1) = 6 \Rightarrow 2x = 7 \Rightarrow x=\tfrac{7}{2}$ (not integer). Case 2: $x < 1 \Rightarrow |x|=-x,\ |1-x|=1-x$ $\Rightarrow -x + (1-x) = 6 \Rightarrow -2x = 5 \Rightarrow x=-\tfrac{5}{2}$ (not integer). Hence, there are **no** non-zero integral solutions.

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