Answer
Verified
424.5k+ views
Hint: In order to solve this problem one should know that the greatest integer function is a piecewise defined function. If the number is an integer, use that integer. If the number is not an integer, use the next smaller integer. And by knowing if we are opening a modulus function then the terms on the other side will be multiplied by $ \pm $. By using these properties you will surely get the right answer.
Complete step-by-step answer:
The given equation is ∣[x]−2x∣=4 where [x] is the greatest integer $ \leqslant $x.
∣[x]−2x∣=4
We will open modulus from the LHS of the equation then there will be $ \pm 4$ in RHS of the equation.
⇒[x] − 2x = ±4 ……(1)
We know that if a number ‘n’ is in greatest integer function [] such that: [n]
Then n = n + {n} where n is an integer part of ‘n’ and {n} is the part whose value belongs to [0,1). So {n} can be 0 but less than 1 and cannot be less than 0.
So, we can say x = x +{x} and [x] = x ({x}=0 when x is an integer)
On putting the value of x and [x] in the equation (1) we get the new equation as,
$ \Rightarrow $x -2 {x} -2x = ±4
$ \Rightarrow $x-2{x}-2x= ±4
$ \Rightarrow $2{x}-x = ±4 (2)
So we will now consider two cases:
Case 1: When x is an integer.
Then {x} = 0
So, x = $ \mp 4$
Then the value of x will be -4, +4
Case 2: When x is not an integer.
Then {x} $ \in (0,1)$.
So, the equation (2) becomes
x = ±4 - 2{x}
In this case x will be an integer only when x is $\dfrac{1}{2}$
Then the equation becomes,
x = $ \pm $4 - 2$\left( {\dfrac{1}{2}} \right)$
x = $ \pm $4 – 1
x = 3, -5
The possible values of x are 3, -5, -4, 4.
Note: Whenever you face such types of problems you have to use the concepts of greatest integer function and modulus function. The greatest integer function is a piecewise defined function. If the number is an integer, use that integer. If the number is not an integer, use the next smaller integer. And by knowing if we are opening a modulus function then the terms on the other side will be multiplied by $ \pm $. By using these concepts you will get the correct solution.
Complete step-by-step answer:
The given equation is ∣[x]−2x∣=4 where [x] is the greatest integer $ \leqslant $x.
∣[x]−2x∣=4
We will open modulus from the LHS of the equation then there will be $ \pm 4$ in RHS of the equation.
⇒[x] − 2x = ±4 ……(1)
We know that if a number ‘n’ is in greatest integer function [] such that: [n]
Then n = n + {n} where n is an integer part of ‘n’ and {n} is the part whose value belongs to [0,1). So {n} can be 0 but less than 1 and cannot be less than 0.
So, we can say x = x +{x} and [x] = x ({x}=0 when x is an integer)
On putting the value of x and [x] in the equation (1) we get the new equation as,
$ \Rightarrow $x -2 {x} -2x = ±4
$ \Rightarrow $x-2{x}-2x= ±4
$ \Rightarrow $2{x}-x = ±4 (2)
So we will now consider two cases:
Case 1: When x is an integer.
Then {x} = 0
So, x = $ \mp 4$
Then the value of x will be -4, +4
Case 2: When x is not an integer.
Then {x} $ \in (0,1)$.
So, the equation (2) becomes
x = ±4 - 2{x}
In this case x will be an integer only when x is $\dfrac{1}{2}$
Then the equation becomes,
x = $ \pm $4 - 2$\left( {\dfrac{1}{2}} \right)$
x = $ \pm $4 – 1
x = 3, -5
The possible values of x are 3, -5, -4, 4.
Note: Whenever you face such types of problems you have to use the concepts of greatest integer function and modulus function. The greatest integer function is a piecewise defined function. If the number is an integer, use that integer. If the number is not an integer, use the next smaller integer. And by knowing if we are opening a modulus function then the terms on the other side will be multiplied by $ \pm $. By using these concepts you will get the correct solution.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The cell wall of prokaryotes are made up of a Cellulose class 9 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
a Tabulate the differences in the characteristics of class 12 chemistry CBSE