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Question
Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.
|x − 4| + |x − 2| = 3
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Solution
|x − 4| + |x − 2| = 3 ...(i)
Case I : x < 2
Equation (i) reduces to
4 − x + 2 − x = 3 ...`[(x < 2 < 4),(therefore x - 4 < 0"," x - 2 < 0)]`
∴ 6 − 3 = 2x
∴ x = `3/2`
Case II : 2 ≤ x < 4
Equation (i) reduces to
4 − x + 2 − x = 3
∴ 2 = 3 (absurd)
∴ There is no solution in [2, 4)
Case III : x ≥ 4
Equation (i) reduces to x – 4 + x – 2 = 3
∴ 2x = 6 + 3 = 9
∴ x = `9/2`
∴ x = `3/2, 9/2` are solutions
The solution set = `{3/2, 9/2}`
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