Question

How many real numbers satisfy the relation [x] = $\frac{3}{2}$ {x}.

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Answer 1

We can write the given relation as {x} = $\frac{2}{3}$ [x]
We know 0 $\le$ {x} $\le$ 1. [x] takes only integer values.

So the values [x] can assume are 0 and 1. For any other value of [x],

.{x} = $\frac{2}{3}$ [x] goes out of its range([0, 1)).

When [x] = 0, {x} = 0

x = [x] + {x}

= 0 + 0 = 0

When [x] = 1, {x} = $\frac{2}{3}$
x = [x] + {x}
= 1 + $\frac{2}{3}$
= $\frac{5}{3}$
So the two values of x we got are 0 and $\frac{5}{3}$