solving a equation (floor function)

Question

I am trying to solve the following problem:

For what real numbers x is: ⌊2x⌋=4⌊x⌋+3?

I'm not sure how to deal with the floor functions, so I have no idea where to start. If someone could walk me through the process that would great!

Answer 1

Notice that $\lfloor 2x\rfloor=2\lfloor x\rfloor$ or $\lfloor 2x\rfloor=2\lfloor x\rfloor+1$ depending on the fractional part. Plugging in the equation, we get

$$\lfloor x\rfloor=-\frac32$$ which is not possible or

$$\lfloor x\rfloor=-1.$$

Hence,

$$-\frac12\le x<0.$$