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Solve for x, where [.] is the greatest integer function and {.} the fractional part: (i) 3[x] - 2x = {x} + 1/2 (ii) [x - 1] + 6x = 3{x} + 2[x] + 6

1. (i) x = -1/2; (ii) x = 7/3
2. (i) x = 1/2; (ii) x = 5/3
3. (i) no solution; (ii) x = 2
4. (i) x = 3/2; (ii) x = 7/3

Correct answer: (i) x = -1/2; (ii) x = 7/3

Solution

Substituting x = n + f (n = [x], f = {x} in [0,1)) turns each equation into a relation between n and f. Requiring n integer and 0 <= f < 1 pins down the unique solution in each case.

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