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

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

Note by Priyankar Kumar
5 years, 1 month ago

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

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

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

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

- 5 years, 1 month ago

Thanks

- 5 years, 1 month 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.

- 5 years, 1 month ago

Thanks.

- 5 years, 1 month ago