Sets, Relation and Function
116863
The number of integral solutions of the equation \(\{x+1\}+2 x=4[x+1]-6\), is
Explanation:
B Given, \(\{\mathrm{x}+1\}+2 \mathrm{x}=4[\mathrm{x}+1]-6\)
We know -
\(\mathrm{x}=\{\mathrm{x}\}+[\mathrm{x}]\)
\(\{\mathrm{x}\}=\mathrm{x}-[\mathrm{x}]\)
Then,
\(x+1-[x+1]+2 x=4[x+1]-6\)
\(3 x+1=5[x+1]-6\)
\(3 x=5\{[x]+1\}-6-1\)
\(3 x=5[x]+5-7\)
\(3 x=5[x]-2\)
Again we put, \(x=\{x\}+[x]\)
\(3\{[\mathrm{x}]+\{\mathrm{x}\}\}=5[\mathrm{x}]-2\)
\(3\{\mathrm{x}\}=2[\mathrm{x}]-2\)
Since,
\(0 \leq\{x\}\lt 1\)
\(0 \leq 3\{x\}\lt 3\)
And
\(0 \leq 2[\mathrm{x}]-2\lt 3\)
\(2 \leq 2[\mathrm{x}]\lt 5\)
\(1 \leq[\mathrm{x}]\lt \frac{5}{2}\)
\(\therefore \quad[\mathrm{x}]=1,2\)
Then, from equation (i), we get -
\({[\mathrm{x}]=1 \Rightarrow \mathrm{x}=1}\)
\({[\mathrm{x}]=2 \Rightarrow \mathrm{x}=\frac{8}{3}}\)So, \(x=1\) is the only integral equation.