- IN ORDER TO ADD OR SUBTRACT RATIONAL EXPRESSIONS,
WE MUST HAVE THE SAME, OR LIKE, DENOMINATORS.
IN THESE TWO EXAMPLES,
NOTICE THE DENOMINATORS LOOK ALMOST THE SAME
BUT NOT QUITE THE SAME.
NOTICE HERE WE HAVE A +X.
HERE WE HAVE A -X.
HERE WE HAVE A NEGATIVE, HERE WE HAVE A +2.
SO THESE DENOMINATORS ARE ACTUALLY OPPOSITES.
SO TO DEAL WITH OPPOSITE DENOMINATORS,
WE CAN FACTOR OUT A NEGATIVE, OR -1,
AND FROM ONE OF THE DENOMINATORS
TO MAKE THE DENOMINATORS LOOK ALMOST THE SAME.
SO WE'LL LEAVE THIS FIRST FRACTION THE SAME,
BUT WE ARE GOING TO INCLUDE THE NUMERATOR AND DENOMINATOR
IN PARENTHESES.
SO WE'LL HAVE THE QUANTITY (12X + 1)
ALL OVER THE QUANTITY (X - 2) PLUS--
HERE WE HAVE THE QUANTITY 7X + 3.
AND NOW HERE WE'RE GOING TO FACTOR OUT A NEGATIVE OR -1.
I'M GOING TO USE A NEGATIVE,
BUT IT'S JUST GOING TO CHANGE THE SIGN OF THESE TWO TERMS.
SO INSTEAD OF HAVING A -X THIS WILL BE A +X,
AND INSTEAD OF BEING A +2 THIS WILL BE -2.
NOTICE IF WE DISTRIBUTE THE NEGATIVE,
WE WOULD HAVE THE SAME EXPRESSION,
THOUGH THE ORDER WOULD BE DIFFERENT.
NOW LET'S TALK ABOUT NEGATIVE FRACTIONS FOR A MOMENT.
IF I HAD, FOR EXAMPLE, -2/3,
THIS NEGATIVE SIGN CAN BE OUT IN FRONT OF THE FRACTION,
IT CAN BE IN THE NUMERATOR, OR IT CAN BE IN THE DENOMINATOR.
ALL OF THESE ARE EQUIVALENT,
SO WHAT WE'LL DO NOW IS JUST MOVE THIS NEGATIVE SIGN
UP INTO THE NUMERATOR,
AND WHEN WE DO THIS, NOTICE HOW THEN--
THE DENOMINATORS WILL BE THE SAME.
SO WE'LL HAVE THE QUANTITY (12X + 1)
ALL OVER THE QUANTITY (X - 2) PLUS--
OUR DENOMINATOR IS NOW GOING TO BE THE QUANTITY (X - 2),
AND THE NUMERATOR IS NOW GOING TO BE -(7X + 3).
NOW THAT WE HAVE LIKE DENOMINATORS,
THE DENOMINATOR IS GOING TO STAY THE SAME,
AND THEN WE'LL ADD THE NUMERATORS.
WE ARE GOING TO CLEAR THE PARENTHESES,
SO NOW WE CAN THINK OF DISTRIBUTING A +1 HERE
AND DISTRIBUTING A -1 HERE.
SO WE'D HAVE 12X + 1.
THIS IS GOING TO BE + A -7X OR -7X.
THIS WILL BE + A -3 OR - 3.
NOW, WE'LL COMBINE THE LIKE TERMS IN THE NUMERATOR.
WE STILL HAVE THE QUANTITY (X - 2) IN THE DENOMINATOR.
NOTICE OUR NUMERATOR IS GOING TO BE 12X - 7X.
THAT'LL BE 5X
AND 1 - 3 IS EQUAL TO -2,
SO WE HAVE 5X - 2 IN OUR NUMERATOR.
NOW, WE DO WANT TO TRY TO SIMPLIFY THIS FRACTION,
BUT SINCE A NUMERATOR DOES NOT FACTOR, THIS DOES NOT SIMPLIFY.
WE CANNOT SIMPLIFY THESE -2'S, FOR EXAMPLE,
BECAUSE WE CANNOT SIMPLIFY ACROSS ADDITION OR SUBTRACTION.
BUT SINCE THE NUMERATOR AND DENOMINATOR
DO ONLY CONTAIN ONE FACTOR,
THE PARENTHESES ARE OPTIONAL.
WE COULD WRITE THIS AS 5X - 2 ALL OVER X - 2.
EITHER OF THESE TWO FORMS ARE ACCEPTABLE.
NOW LET'S TAKE A LOOK AT OUR SECOND EXAMPLE
WHEN WE HAVE SUBTRACTION.
AGAIN, NOTICE HOW THE DENOMINATORS ARE OPPOSITES.
SO FOR THE FIRST STEP,
WE'RE GOING TO INCLUDE PARENTHESES EVERYWHERE.
SO WE'LL HAVE THE QUANTITY (X + 1)
OVER THE QUANTITY (X - 5)
MINUS THE QUANTITY (2X + 14) OVER--
AND AGAIN, BECAUSE OUR DENOMINATORS ARE OPPOSITES,
HERE WE'RE GOING TO FACTOR OUT A NEGATIVE,
AND THIS WILL BE A +X AND A -5.
NOW REMEMBER, THIS JUST MEANS THIS FRACTION IS NEGATIVE.
AND REMEMBER, SUBTRACTING A NEGATIVE
IS THE SAME AS ADDING A POSITIVE.
SO NOW WE CAN REWRITE THIS AS AN ADDITION PROBLEM.
WE WOULD HAVE THE QUANTITY (X + 1)
ALL OVER THE QUANTITY (X - 5) PLUS THE QUANTITY (2X + 14)
ALL OVER THE QUANTITY (X - 5).
AND NOW THAT WE HAVE A COMMON DENOMINATOR,
WE CAN ADD THE NUMERATORS.
AND AT THE SAME TIME WE'LL CLEAR THE PARENTHESES
WHICH WE CAN THINK OF JUST DISTRIBUTING A +1,
BUT IT'S NOT GOING TO CHANGE ANY SIGNS.
SO HERE WE'D HAVE (X + 1) + (2X +14).
NOW, WE'LL COMBINE LIKE TERMS.
1X + 2X IS 3X.
1 + 14 IS 15, SO + 15.
NOTICE HOW THE NUMERATOR DOES FACTOR.
THERE'S A COMMON FACTOR OF 3 HERE,
SO WE'D HAVE 3 TIMES THE QUANTITY (X + 5)
ALL OVER THE QUANTITY (X - 5).
NOTICE HOW THESE FACTORS ARE NOT THE SAME.
ONE IS A SUM, AND ONE IS A DIFFERENCE.
SO THIS DOES NOT SIMPLIFY.
SO WE'D LEAVE OUR ANSWER IN THIS FORM.
OKAY, I HOPE YOU FOUND THIS HELPFUL.