Physics calculation- speed of a star?

Relative to what?

its moving away from the earth…because the wavelenght increases! When I did the calculation I got the correct answer but I cant figure out if the way that I solved it is acceptable! I did this in a different way but i cant remember the method that I used!!

This problem is discussing doppler shift.
Here’s simple way to look at it. First remember the equation c=f Lamda The frequency x the wavelength is equal to the speed of light.
Ignoring relativity, if the star was moving away from you at half the speed of light the wavelength is stretched out by the same ratio. If it is moving toward you the wave fonts are squeezed together and the wavelength is shorter.
If the wavelength is stretched out by a certain % that is the same number as the % speed of light. So, you are getting 590nm instead of 587nm. That is a difference of 3nm. (3nm/587nm) x c is the star velocity away from you. This works for sound waves too. You have to replace c with M for speed of sound.

Bonus question: You just received a ticket for running a red light and must prepare a defense in court. You say the light looked green to you. How fast do you have to be driving to doppler shift a red light 640 nm to look green 510 nm? Hint. It would be one hell of a speeding ticket.

The doppler shift is a powerful tool.
That’s how astronomers determine if a planet is orbiting a distant star. They look at the change in the wavelength of the Hydrogen alpha line (or beta or other) of the star and see if it varies periodically. If it does, that means the star is moving towards us and away from us in response to a another mass nearby. By knowing the mass of the star and measuring the frequency of the shift they can determine the mass of the planet and its distance from the star.
Police radar and lidar works the same way.