Question

If $y+5\left[x\right]=1\&2[x+2]+7=\left[x\right],$ then the value of $[\frac{y}{2}+x]=$, where $[.]$ represents the greatest integer function.

Answer 1

Given: $y+5\left[x\right]=1$ and $2[x+2]+7=\left[x\right]$

Applying the property, $[x\pm n]=\left[x\right]\pm n,n\in Z.$
$\Rightarrow [x+2]=\left[x\right]+2$

Substituting this in the above equation we get,
$2\left(\right[x]+2)+7=\left[x\right]$
$\Rightarrow 2\left[x\right]+4+7=\left[x\right]$
$\Rightarrow \left[x\right]=-11$
$\Rightarrow \left[x\right]\le x<\left[x\right]+\mathrm{1...}\left(a\right)$
$\Rightarrow -11\le x<-11+1$
$\Rightarrow -11\le x<-10$

Substituting this in the given equation we get,
$y+5\left[x\right]=1$
$\Rightarrow y+5(-11)=1$
$\Rightarrow y=1+55=56$
$\Rightarrow \frac{y}{2}=\frac{56}{2}=28$

Adding $\frac{y}{2}$ to the above inequality we get,
$\Rightarrow -11+28\le \frac{y}{2}+x<-10+28$
$\Rightarrow 17\le \frac{y}{2}+x<18$

By comparing with the relation (a),
$[\frac{y}{2}+x]=17$