Does finding real income adjust for inflation? I have been reading from this textbook which talks about CPI and inflation 

http://faculty.washington.edu/cnelson/Chap04.pdf

On pages 5 and 6, it talks about how they are deflating the nominal income using the CPI. But in page 6, they work out the percentage difference in real income between two years, and they state that is the net difference of her increase in real income and inflation

But I thought using the CPI was getting rid of the effect of inflation? What am I not undersranding?
Indeed deflating nominal income using CPI does get rid of the effect of inflation. I think you're just getting confused by the phrasing.

I assume you are referring to the following sentence on pg. 6:


  We  have  shown  that  the  net  result  of  her  raise  and  inflation  was  an  increase of 7.9% in real income.


Let me rephrase this sentence: we have shown that after raising her salary by 10% and then adjusting for inflation, the net effect is that her real income increased by 7.9%.