Answer
452.7k+ 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
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)