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

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 thatx = \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.