Question

Show that [x] + [x + 1/2] = [2x] when x is a real number.

[x] is the floor function, where [5/2] = 2 and [-5/2] = -3.

Answer 1

For positive values of x, there are 3 cases to consider
1. x is an integer.
e.g. x = 6
LS = [6] + [6+.5]
= 6 + 6
= 12
RS = [2x6] = [12] = 12 = LS

2. x is a decimal with a decimal value < .5
e.g. x = 3.4
LS = [3.4] + [3.4 + .5]
= 3 + 3
= 6
RS = [2 x 3.4]
= [ 6.8]
= 6 = LS

3. x is a decimal number whose decimal is ≥ .5

e.g. x = 8.75
LS = [8.75] + [8.75 + .5]
= 8 + 9
= 17
RS = [2 x 8.75]
= [17.5]
= 17 = LS

I will leave it up to you to "show" it to be true for negative values of x

Can you see why adding .5 to a decimal of case #1 will not bring it to the next whole number, nor will doubling the number, but
if the decimal is ≥ .5 then adding .5 or doubling the number will drag it into the next whole number.

The question said to "show", it did not say "prove", so an explanation should do it.