**12. Show that [x] + [x + 1/2] = [2x] whenever x is a real number.**

Solution: We have to prove the following equation

[x]+[x+1/2]=[2x]….(1)

Where x is a real number

Let us consider the LHS of the above equation.

We define

x=[x]+{x}….(2)

Where [x] is the integral part and {x} is the fractional part of x

Now, substituting (2) in (1) will give us

LHS=[[x]+{x}]+[[x]+{x}+1/2]

=[x]+[{x}]+[x]+[{x}+1/2]

We have used the fact that [[x]]=[x] as [x] is itself an integer.

Proceeding forward gives us

LHS=2[x]+[{x}]+[{x}+1/2]

=2[x]+0+[{x}+1/2]

This is because the greatest integer value of any fractional value is zero.

Consider two following cases.

**Case 1**

We assume that [{x}+1/2]=0

So,

LHS=2[x]+0

=2[x]

=[2x]

This is because 2 is an integer value and can be including in the greatest integer part.

**Case 2**

We assume that [{x}+1/2]=1

So,

LHS=2[x]+1

=2[x]+[2{x}]

=[[2x]]+[2{x}]

=[2[x]+2{x}]

Therefore,

LHS=[2([x]+{x})]

=[2x]

This we get from (2)

Hence, we have proved the equation (1)