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Using [x] for the greatest integer function, determine the set of x for each: (i) [x] = 10, (ii) [x]² - [x] - 2 = 0, (iii) [x] >= 5, (iv) [x] < -3. Choose the correct combined answer.

1. (i) [10,11) (ii) [-1,0) U [2,3) (iii) [5,infinity) (iv) (-infinity,-3)
2. (i) (10,11) (ii) [-1,0) U [2,3) (iii) (5,infinity) (iv) (-infinity,-3]
3. (i) [10,11) (ii) [2,3) (iii) [5,infinity) (iv) (-infinity,-4)
4. (i) [10,11] (ii) [-1,0] U [2,3] (iii) [5,infinity) (iv) (-infinity,-3)

Correct answer: (i) [10,11) (ii) [-1,0) U [2,3) (iii) [5,infinity) (iv) (-infinity,-3)

Solution

(i) [x]=10 means 10 <= x < 11 -> [10,11). (ii) [x]²-[x]-2 = ([x]-2)([x]+1) = 0 -> [x]=2 or [x]=-1, giving x in [2,3) U [-1,0). (iii) [x] >= 5 means x >= 5 -> [5,infinity). (iv) [x] < -3 means [x] <= -4 (integer), so x < -3 -> (-infinity,-3).

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