Can someone show me how they used the Chinese Remainder Theorem on the example in the slides?

I think there is a mistake.

Equations should be x = +1(mod 7), x = -1(mod 7) AND  

                              x = +3(mod 11), x = -3(mod 11)

Then pick each set and apply CRT, you will get 4 equations (+-6) and (+-27)

ex:       Pick Set 1:  x = +1(mod 7)     x = +3(mod 11)  -- Apply CRT

    then Pick  Set 2: x = +1(mod 7)     x = -3(mod 11)  -- Apply CRT

I hope it will help you..!!