- TO ADD OR SUBTRACT RATIONAL EXPRESSIONS
WE MUST HAVE THE SAME OR LIKE DENOMINATORS.
NOTICE THESE TWO EXAMPLES DO NOT CONTAIN LIKE DENOMINATORS.
SO FOR THE FIRST STEP, WE'RE GOING TO RE-WRITE THE PROBLEM
LEAVING A LOT MORE SPACE HORIZONTALLY
AND INCLUDING PARENTHESES AROUND THE DENOMINATORS.
SO WE'LL HAVE 3 OVER THE QUANTITY (X - 3) + 2
OVER THE QUANTITY X + 5.
WELL TO HAVE LIKE DENOMINATORS OR THE SAME DENOMINATORS
BOTH DENOMINATORS MUST CONTAIN THE SAME FACTORS.
NOTICE HOW THIS DENOMINATOR HAS A FACTOR OF X + 5
BUT THIS ONE DOESN'T.
SO WE CAN MULTIPLY THIS DENOMINATOR
BY A FACTOR OF X + 5
AS LONG AS WE DO THE SAME TO THE NUMERATOR.
REMEMBER ANYTHING OVER ITSELF IS EQUAL TO 1.
SO THIS IS LIKE MULTIPLYING BY 1.
NOTICE THIS DENOMINATOR DOES NOT HAVE THE FACTOR OF X - 3
SO AGAIN, WE CAN MULTIPLY THIS BY X - 3
AS LONG AS WE DO THE SAME TO THE NUMERATOR.
AND NOW NOTICE THE DENOMINATORS
CONTAIN THE SAME FACTORS
AND THEREFORE WE HAVE THE SAME DENOMINATORS.
SO NOW WE CAN ADD THESE FRACTIONS
BY KEEPING THE DENOMINATOR THE SAME
AND ADDING THE NUMERATORS.
SO OUR DENOMINATOR IS GOING TO BE THE QUANTITY (X - 3)
x (THE QUANTITY X + 5).
REMEMBER THE ORDER OF THIS DOESN'T MATTER
BECAUSE IT'S MULTIPLICATION.
AND NOW WE'RE GOING TO COMBINE THE NUMERATORS
SO WE'LL HAVE (3 x THE QUANTITY X) + (5 + 2)
x (THE QUANTITY X - 3).
SO NOW WE'LL DISTRIBUTE IN THE NUMERATOR
AND COMBINE LIKE TERMS.
SO HERE WE'LL HAVE 3X + 15, HERE WE'LL HAVE 2X - 6.
AND NOW WE'LL COMBINE LIKE TERMS IN THE NUMERATOR.
WE HAVE 3X + 2X THAT'S 5X.
AND THEN WE HAVE 15 - 6 THAT'S EQUAL TO 9.
AND SINCE 5X + 9 DOES NOT FACTOR THIS DOES NOT SIMPLIFY
SO WE COULD LEAVE OUR ANSWER LIKE THIS
OR SOMETIMES YOU'LL SEE IT
WHERE WE HAVE PARENTHESES IN THE NUMERATOR AS WELL.
EITHER FORM'S ACCEPTABLE.
NOW LET'S TAKE A LOOK AT AN EXAMPLE
OF A SUBTRACTION PROBLEM.
AGAIN, FOR THE FIRST STEP,
WE'RE GOING TO LEAVE SOME ROOM HORIZONTALLY
AND PUT PARENTHESES AROUND THE DENOMINATORS.
SO WE'LL HAVE
(7 OVER THE QUANTITY X + 2) - (5 OVER THE QUANTITY X + 3).
AND NOW WE WANT TO HAVE DENOMINATORS
WITH THE SAME FACTORS.
SO THIS DENOMINATOR NEEDS A FACTOR OF X + 3.
SO WE'LL MULTIPLY THIS FRACTION BY X + 3/X + 3
AND THIS DENOMINATOR NEEDS A FACTOR OF X + 2.
SO WE'LL MULTIPLY BY X + 2/X + 2.
AND NOW THAT WE HAVE THE SAME DENOMINATOR
THE DENOMINATOR'S GOING TO STAY THE SAME.
QUANTITY X + 2 x THE X + 3
AND NOW WE'LL COMBINE THE NUMERATORS BY SUBTRACTING.
SO WE'LL HAVE
(7 x THE QUANTITY X + 3 - 5 x THE QUANTITY X + 2.
NOW WE'LL CLEAR THE PARENTHESES
AND COMBINE THE LIKE TERMS IN THE NUMERATOR,
SO HERE WE'LL DISTRIBUTE 7.
SO WE'LL HAVE 7X + 21
AND HERE, BECAUSE OF THE SUBTRACTION
WE'LL THINK OF DISTRIBUTING A -5.
SO WE'LL HAVE -5X OR - 5X AND THEN -10 OR - 10.
SO AGAIN, WE'LL HAVE (7X + 21) - (5X - 10).
AND NOW WE'LL COMBINE THE LIKE TERMS IN THE NUMERATOR.
7X - 5X WOULD BE 2X AND 21 -10 IS EQUAL TO 11 SO WE HAVE +11.
AGAIN, THE NUMERATOR DOES NOT FACTOR
AND THEREFORE THIS DOES NOT SIMPLIFY
AND AGAIN, SOMETIMES YOU WILL SEE THE NUMERATOR
IN PARENTHESES AS WELL.
AGAIN, EITHER FORM'S ACCEPTABLE.
OKAY. I HOPE YOU FOUND THESE TWO EXAMPLES HELPFUL.
WE'LL TAKE A LOOK AT TWO MORE IN THE NEXT VIDEO.