Question

Find the number of solutions for $4\left\{x\right\}=x+\left[x\right]$, where $\{.\}$, [.] represents fractional part and greatest integer function.

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Answer 1

As we know ,to find number of solutions of two curves we should find the point of intersection

of two curves.

$\therefore$ $4\left\{x\right\}=x+\left[x\right]$

$\Rightarrow 4(x-[x\left]\right)=x+\left[x\right]$ $\{\because x=[x]+\{x\left\}\right\}$

$\Rightarrow 4x-x=4\left[x\right]+\left[x\right]$

$\Rightarrow 3x=5\left[x\right]$

$\Rightarrow \left[x\right]=\frac{3}{5}x$ ...(i)

Clearly ,the two graphs intersects when [x]=0 and

[x]=1 ...(ii)

$\therefore x=\frac{5}{3}\left[x\right]$ [From Equation .(i) and (ii)

$x=\frac{5}{3}$ and $x=\frac{5}{3}$ (1)

$\therefore x=0andx=\frac{5}{3}$ are the only two solutions.