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solakis1
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Solve the following equation:
$[x]=2x+1$,where [x] is the floor function
$[x]=2x+1$,where [x] is the floor function
... or 0.Country Boy said:2r= -[x]- 1. Since -[x]-1 is an integer, r must be 1/2 or -1/2.
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.solakis said:Solve the following equation:
$[x]=2x+1$,where [x] is the floor function
https://www.desmos.com/calculator/0cpj2izgfiDaalChawal 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 :)
Typo perhaps?solakis said:... Which is equivalent to :
x=-1 or x= -(4/3) ...
Other than that, your solution looks fine. :Dsolakis said:The whole problem is a typo I solved the equation [x]= 3x+2 instead the one which was in OP ,[x]=2x+1
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.
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.
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.
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.
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.