I would never disagree with IRonMaN!

Tell the client that they are allowed to take a tax credit on the Illinois tax return for ALL of the taxes that they paid Texas. 

P.S.

You done good the first time. 

P.P.S. If all else fails, share this link with the client https://letmegooglethat.com/?q=does+an+illinnois+resident+have+to+pay+taxes+on+income+earned+in+texa...