MHB Solving the Equation: x+[2x]+[3x]=7

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The equation x+[2x]+[3x]=7 is discussed as an easy problem, prompting questions about its posting. Participants categorize real numbers into various sets based on whether they are integers or non-integers and their relationships with 2x and 3x. A reference to Desmos suggests that the equation appears to imply x = 4/3, but this is incorrect as it does not satisfy the equation. The correct interpretation indicates that 4/3 leads to a left-hand side of 8, not matching the right-hand side of 7. The conversation emphasizes the importance of careful analysis in solving such equations.
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An easy one:

x+[2x]+[3x]=7
 
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Yes, that is an easy problem! Why did you post it?

I would start by dividing the real numbers into classes:
1) The set of integers.
2) The set of non-integer, x, such that 2x is an integer
3) The set of non-integers, x, such that 2x is not an integer but 3x is.
4) The set of non-integers, x, such that neither 2x nor 3x is an integer but 6x is.
5) The set all other real numbers.
 
Because RHS is integer so LHS is integer

As $\lfloor 2x \rfloor$ and $\lfloor 3x \rfloor$ are integers so x is integer so $\lfloor 2x \rfloor = 2x $ and $\lfloor 3x \rfloor = 3x$

so x + 2x + 3x = 6x = 7 so $x = \frac{7}{6}$ which is not integer so NO solution
 
Beer soaked ramblings follow.
solakis said:
An easy one:

x+[2x]+[3x]=7
Desmos somehow gives the impression that $x + \lfloor 2x \rfloor + \lfloor 3x \rfloor = 7$ is basically the line $x = \frac{4}{3}$.
https://www.desmos.com/calculator/stz3o2wn2h
 
The answer given by Kaliprasad is the right one 7\6 is not an integer
4/3 is not a solution of the above equation LHS is 8 RHS Is 7
 
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