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

  • MHB
  • Thread starter solakis1
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In summary, the conversation discusses a problem involving the equation x+[2x]+[3x]=7 and different sets of real numbers that can be divided into based on their properties. The conclusion is that the correct solution is x=4/3 and not 7/6 as previously suggested.
  • #1
solakis1
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An easy one:

x+[2x]+[3x]=7
 
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  • #2
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.
 
  • #3
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
 
  • #4
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
 
  • #5
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
 
Last edited:

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

What is the value of x in the equation x+[2x]+[3x]=7?

The value of x in this equation is not a specific number, but rather a variable that can take on different values depending on the context of the problem. In order to solve for x, we need more information or additional equations.

How do I solve this equation?

To solve this equation, you can use algebraic methods such as combining like terms, distributing, and isolating the variable on one side of the equation. It may also be helpful to use a calculator or graphing software to visualize the equation and find possible solutions.

Can this equation have multiple solutions?

Yes, this equation can have multiple solutions. Since there is only one variable, there can be an infinite number of values that satisfy this equation. However, in order to find a specific solution, we need more information or additional equations.

Is there a specific method for solving equations with multiple variables?

Yes, there are specific methods for solving equations with multiple variables. One common method is substitution, where one variable is isolated and substituted into another equation to solve for the other variable. Another method is elimination, where equations are combined and variables are eliminated to solve for the remaining variable.

Can this equation be solved without using algebra?

It is possible to solve this equation without using algebra, but it may be more difficult and time-consuming. Graphing or using a calculator can help find approximate solutions, but algebraic methods are typically more accurate and efficient in solving equations.

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