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  4. Number of solutions of the equation 3 x+ x =4[x] will be (where [] represent greatest integer and fractional part respectively)

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Q. Number of solutions of the equation $3 x+\{x\}=4[x]$ will be (where [] \& \{ \} represent greatest integer and fractional part respectively)

Relations and Functions - Part 2

Solution:

$ \Theta 3[x]+3\{x\}+\{x\}=4[x] $
$\Rightarrow[ x ]=4\{ x \} $
$\Theta 0 \leq\{x\}<1 \Rightarrow 0 \leq[x]<4 $
$\Rightarrow[ x ]=0,1,2,3 $
$\Rightarrow\{x\}=0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4} $
$\Rightarrow x =0, \frac{5}{4}, \frac{10}{4}, \frac{15}{4} $
$\therefore \text { Number of solutions }=4 $