Question

If $\left[x\right]=2,\left[y\right]=-5$ then find the range of $[x+y+1]$ where $[.]$ represents greatest integer function.

Answer 1

Given: $\left[x\right]=2,\left[y\right]=-5$
Let $Z=[x+y+1]$.

As we know $[x+y+1]=[x+y]+1$

Now, using the inequality:
$\left[x\right]+\left[y\right]\le [x+y]<\left[x\right]+\left[y\right]+1$
Add 1 one each sides, we get
$\Rightarrow \left[x\right]+\left[y\right]+1\le [x+y]+1<\left[x\right]+\left[y\right]+2$
$\Rightarrow \left[x\right]+\left[y\right]+1\le [x+y+1]<\left[x\right]+\left[y\right]+2$
$\Rightarrow \left[x\right]+\left[y\right]+1\le Z<\left[x\right]+\left[y\right]+2$
​$\Rightarrow 2-5+1\le Z<2-5+2$
​$\Rightarrow -2\le Z<-1$
$\Rightarrow [x+y+1]\in [-2,-1]$
But since [x+y+1] is always an interger
$\therefore [x+y+1]=\{-2,-1\}$
Hence the correct answer is Option A.