the number of solutions of [2x]-3{2x}=1

the number of solutions of [2x]-3{2x}=1? How to solve such problems

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

If $\lfloor 2x \rfloor - 3\{2x\} = 1$ then $3\{2x\} = \lfloor 2x\rfloor - 1$ is an integer, so $\{2x\} = 0,1/3,2/3$ .

If $\{2x\} = 0$ then $\lfloor 2x\rfloor - 1 = 0$ , so $\lfloor 2x\rfloor = 1$ , and hence $2x=1$ , so $x=1/2$ .
If $\{2x\} = 1/3$ then $\lfloor 2x\rfloor - 1 = 1$ , so $\lfloor 2x \rfloor = 2$ , and hence $2x=7/3$ , so $x=7/6$ .
If $\{2x\} = 2/3$ then $\lfloor 2x\rfloor - 1 = 2$ , so $\lfloor 2x \rfloor = 3$ , and hence $2x=11/3$ , so $x=11/6$ .

Mark Hennings - 7 years, 10 months ago

Thanks

Priyankar Kumar - 7 years, 10 months ago

(I assume that the square and curly brackets represent floor and fractional part function respectively)

I am not the right person to help out in such problems but the following is worth a try. Add and subtract {2x} in the LHS. The equation now becomes 2x-4{2x}=1 or 2x=1+4{2x}. You can now plot a graph to find the number of solutions.

Pranav Arora - 7 years, 10 months ago

Thanks.

Priyankar Kumar - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$