Question

Find the number of solution for $\left| {\left[ x \right] - 2x} \right| = 4$ where $\left[ x \right]$ is the greatest integer less than or equal to x.
A) 2
B) 4
C) 1
D) Infinite

Answer 1

Hint: In questions involving greatest integer function you can always use fractional part function to simplify the expression and find corresponding solutions. Using $x = \left[ x \right] + \left\{ x \right\}$, we will simplify the expression and then compare LHS and RHS for finding different values of {x}. And will make different cases and in each case find the range of x.

Complete step-by-step answer:
We have, $\left| {\left[ x \right] - 2x} \right| = 4$
We know that$x = \left[ x \right] + \left\{ x \right\}$, where [x] is the greatest integer less than or equal to x and {x} is fractional part of x.
Putting value of x in given expression we get,
\[
  \left| {\left[ x \right] - 2x} \right| = \left| {\left[ x \right] - 2(\left[ x \right] + \left\{ x \right\})} \right| \\
   = \left| {\left[ x \right] - 2\left[ x \right] - 2\left\{ x \right\}} \right| \\
   = \left| { - \left[ x \right] - 2\left\{ x \right\}} \right| \\
\]
We know that $\left| { - x} \right| = x$
So,\[\left| {\left[ x \right] - 2x} \right| = \left| {\left[ x \right] + 2\left\{ x \right\}} \right|\] (Using property of Modulus)
So, we have \[\left| {\left[ x \right] + 2\left\{ x \right\}} \right| = 4\]
Now, RHS = 4 is an integer $ \Rightarrow $ LHS must be an integer$ \Rightarrow $\[\left| {\left[ x \right] + 2\left\{ x \right\}} \right|\] is integer
But since [x] is integer $ \Rightarrow $ 2{x} is also an integer. But since {x} is fractional part so 2{x} can take only integer values i.e. 0 or 1
$ \Rightarrow $2{x} =0 or 2{x} =1
$ \Rightarrow ${x}=0 or {x} =0.5
Case I: {x} =0: Using $x = \left[ x \right] + \left\{ x \right\}$ we get,
     $[x] = x$ And $\{ x\} = 0$
So \[
  \left| {\left[ x \right] + 2\left\{ x \right\}} \right| = \left| x \right| = 4 \\
   \Rightarrow x = + 4, - 4 \\
\]
So, In this case we have two solutions 2 solutions
Case II: If $\left\{ x \right\}{\text{ }} = 0.5 \Rightarrow 2\{ x\} = 2 \times 0.5 = 1$
 $ \left| {2\{ x\} + [x]} \right| = \left| {1 + [x]} \right| = 4 \\
   \Rightarrow 1 + [x] = 4{\text{ or }}1 + [x] = - 4 \\
   \Rightarrow [x] = 3{\text{ or }}[x] = - 5 \\
$
     $
  [x] = 3{\text{ }} \Rightarrow {\text{x}} \in [3,4{\text{)}} \\
  {\text{and }}[x] = - 5 \Rightarrow x \in [ - 5, - 4) \\
$
     So x has infinite solutions.

Option D is the correct answer.

Note: In questions involving a number of solutions have to consider all possible conditions that can arise. You have to take care of all the cases and solve them to get the values of x.If a number, say x is such that $1 \leqslant x \leqslant 2$ then x can take any value between 1 to 2 i.e. it will have infinite number of solutions.