Solving the Floor Function Equation: $[x]=2x+1$

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In summary, the solutions to the equation $[x] = 2x+1$, where $[x]$ is the floor function, are $x = -1$ and $x = -\frac{3}{2}$.
  • #1
solakis1
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Solve the following equation:

$[x]=2x+1$,where [x] is the floor function
 
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  • #2
We can write x as [x}+ r where r is the "fraction part". The equation is [x]= 2[x]+ 2r+ 1 so r2+ 1= -[x]. 2r= -[x]- 1. Since -[x]-1 is an integer, r must be 1/2 or -1/2.
 
  • #3
So what are the values for x??
 
  • #4
Country Boy said:
2r= -[x]- 1. Since -[x]-1 is an integer, r must be 1/2 or -1/2.
... or 0.
 
  • #5
So what are the values of x
 
  • #6
Is the answer x belongs to [-1 , 0 ) ?
 
  • #7
solakis said:
Solve the following equation:

$[x]=2x+1$,where [x] is the floor function
Let $x = n+r$, where $n = \lfloor x\rfloor$ is the integer part of $x$, and $r$ is the fractional part with $0\leqslant r<1$. Then the equation $\lfloor x \rfloor = 2x+1$ becomes $n = 2(n+r)+1$, so that $n = -2r-1$. The right side of that equation is negative, therefore $n$ must be negative. But $n$ is an integer, which means that $r$ must be $0$ or $\frac12$. If $r=0$ then $n = -1$, and if $r = \frac12$ then $n=-2$. The two solutions for $x$ are therefore $x=-1$ and $x = -1.5$. Those are the only solutions.
 
  • #8
You're right thanks I made a silly mistake.This is my approach.
We know that {x} + [x] = x
Now $x-{x} = 2x + 1 $
so $-{x}=x + 1$
Now just draw graph :)
 
  • #9
Beer inspired graph follows.
DaalChawal said:
You're right thanks I made a silly mistake.This is my approach.
We know that {x} + [x] = x
Now $x-{x} = 2x + 1 $
so $-{x}=x + 1$
Now just draw graph :)
https://www.desmos.com/calculator/0cpj2izgfi
 
  • #10
Hey sorry I meant x - {x} = 2x + 1
-{x} = x + 1
It was a typo.
 
  • #11
[sp]Definition of floor value of real no x denoted by [x]

$\forall A\forall B( [A]=B\Leftrightarrow B\leq [A]<B+1 \wedge( B\in Z))$

In the above definition put A=x and B =3x+2 and we have:

[x]=3x+2 $\Leftrightarrow 3x+2\leq x< 3x+3\wedge 3x+2\in Z$ , where Z is the set of integers

Which is equivalent to :

$\frac{-3}{2}<x\leq -1\wedge 3x+2\in Z$

Which is equivalent to :

$\frac{-5}{2}<3x+2\leq -1\wedge 3x+2\in Z$

Which is equivalent to :

3x+2=-1 or 3x+2=-2

Which is equivalent to :

x=-1 or x= -(4/3)

Note since all the steps in the proof are equivalent we can say these are the only solutions to the above equation[/sp]
 
  • #12
Beer induced observation follows.
solakis said:
... Which is equivalent to :

x=-1 or x= -(4/3) ...
Typo perhaps?
 
  • #13
The whole problem is a typo I solved the equation [x]= 3x+2 instead the one which was in OP ,[x]=2x+1
 
  • #14
solakis said:
The whole problem is a typo I solved the equation [x]= 3x+2 instead the one which was in OP ,[x]=2x+1
Other than that, your solution looks fine. :D
 
  • #15
Since i solved the wrong problem let me solve the right one using the n substitution.
S0 let $[x]=n=2x+1 $ this implies that $ n\leq x <n+1$ and $x=\dfrac{n-1}{2}$ and combining the two we have"

$n\leq\dfrac{n-1}{2}<n+1$ which implies $3<n\leq -1$ which implies $n=-1$ or $n=-2$ since n is integer

Hence the original equation gives :$ -1=2x+1$ or $-2=2x+1$ which implies $x=-1$ or $x=-(3/2)$
 

Related to Solving the Floor Function Equation: $[x]=2x+1$

1. What is the floor function equation?

The floor function equation, denoted by [x], is a mathematical function that takes a real number as an input and returns the largest integer less than or equal to that number. In other words, it rounds down the input to the nearest whole number.

2. How do you solve the floor function equation?

To solve the floor function equation [x]=2x+1, you can use the definition of the floor function to set up an inequality. For example, if [x]=5, then 5≤x<6. From there, you can solve for x by isolating it on one side of the inequality. In this case, x would be equal to 5.

3. Can the floor function equation have multiple solutions?

Yes, the floor function equation can have multiple solutions. Since the floor function rounds down to the nearest whole number, there can be multiple real numbers that satisfy the equation. For example, if [x]=3, then both 3 and 3.5 are solutions to the equation 3=2x+1.

4. Are there any special cases when solving the floor function equation?

Yes, there are a few special cases to consider when solving the floor function equation. One is when the input is a negative number. In this case, the floor function will round down to the nearest integer that is less than the input, so the solution may be a negative number. Another special case is when the input is already a whole number, in which case the floor function will return the input itself as the solution.

5. How is the floor function equation used in real life?

The floor function equation has many practical applications in fields such as computer science and engineering. It is often used to round down measurements or values to the nearest whole number, which can be helpful in data analysis and programming. It is also used in mathematical proofs and calculations involving inequalities.

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