Question

The number of solutions of $\left|\right[x]-2x|=4$, where$\left[x\right]$ denotes the greatest integer$\le x$, is

• A. Infinite
• B. $4$
• C. $3$
• D. $2$

Answer 1

The correct option is B

$4$

Find the number of solutions for the given equation

Given, $\left|\right[x]-2x|=4$

We have,

$$\begin{gathered} \left|\right[x]–2x|=4 \\ \Rightarrow \left[x\right]–2x=\pm 4 \\ \Rightarrow \left[x\right]=2x\pm 4 \end{gathered}$$

If$x$ is an integer, then $x=2x\pm 4$

$$\begin{gathered} \Rightarrow x\pm 4=0 \\ \Rightarrow x=\pm 4 \end{gathered}$$

If $x$ is not an integer, then $x=p+q$ where $p$ is an integer and $q$is a fraction.

$$\begin{gathered} \left[x\right]=p \\ =x–q \end{gathered}$$

$$\begin{gathered} \therefore x–q=2x\pm 4 \\ \Rightarrow x=\frac{-q}{4} \end{gathered}$$

So, $4$values of $x$ exists.

Hence, option (B) is the correct answer.