Home
Class 12
MATHS
Let {x} and [x] denotes the fraction fra...

Let `{x}` and `[x]` denotes the fraction fractional and integral part of a real number `(x)`, respectively. Solve `|2x-1|=3[x]+2{x}`.

Text Solution

Verified by Experts

Case I `2x-1ge0` or `xge1/2`
Then given equation convert to
`2x-1=3[x]+2{x}`………
`:'x=[x]+{x}` …..ii
From Eq. I and ii we get
`2([x]+{x})-1=3[x]+2{x}`
`:.[x]=-1`
`:.-1lexlt0`
No solution `[:'x ge1/2]`
Case II `2x-1lt0` or `x lt1/2`
Then given equation reduces to
`1-2x=3[x]+2{x}`.....iii
`:'[x=[x]+{x}`.....iv
From Eqs (iii) and (iv) we get
`1-2([x]+{x})=3[x]+2{x}`
`implies1-5[x]=4{x}`
`:.{x}=(1=5[x])/4`.......v
Now `0le{x}lt1`
`implies0le(1-5[x])/4lt1`
`implies0le1-5[x]gt-4` ltbRgt `implies1ge5[x]ge-3` or `-3/5lt[x]le1/5`
`:.[x]=0`
From Eq. (v) `{x}=1/4`
`:.x=0+1/4=1/4`
Doubtnut Promotions Banner Mobile Dark
|

Similar Questions

Explore conceptually related problems

If [x] is the integral part of a real number x . Then solve [2x]-[x+1]=2x

If {x} and [x] represent fractiona and integral part of x respectively then solve the equation x-1=(x-[x])(x-{x})

If f(x) and [x] denote respectively the fractional and integeral parts of a real number x, then the number of solution of the euation 4{x}=x+[x] , is

If {x} and [x] represent fractional and integral part of x respectively, find the value of [x]+sum_(r=1)^(2000)({x+r})/2000

Given f(x)={3-[cot^(-1)((2x^3-3)/(x^2))]forx >0{x^2}cos(e^(1/x))forx<0 (where {} and [] denotes the fractional part and the integral part functions respectively). Then which of the following statements do/does hold good?

Solve the equation [x]{x}=x, where [] and {} denote the greatest integer function and fractional part, respectively.

Number of points of non-differentiability of the function g(x) = [x^2]{cos^2 4x} + {x^2}[cos^2 4x] +x^2 sin^2 4x + [x^2][cos^2 4x] + {x^2}{cos^2 4x} in (-50, 50) where [x] and {x} denotes the greatest integer function and fractional part function of x respectively, is equal to :

Solve 3x+8 gt2 , when (i) x is an integer. (ii) x is a real number.

If [x] and (x) are the integral part of x and nearest integer to x then solve (x)[x]=1

Consider the function f(x)={{:(x-[x]-(1)/(2),x !in),(0, "x inI):} where [.] denotes the fractional integral function and I is the set of integers. Then find g(x)max.[x^(2),f(x),|x|},-2lexle2.