- TO SOLVE A RATIONAL EQUATION LIKE WE HAVE HERE,
OUR GOAL IS TO CLEAR THE FRACTIONS FROM THE EQUATION
AND THEN SOLVE THE RESULTING EQUATION.
TO DO THIS, WE WANT TO MULTIPLY BOTH SIDES OF THE EQUATION
BY WHAT WOULD BE THE LEAST COMMON DENOMINATOR,
SO THE FIRST STEP IS TO WRITE EACH FRACTION
AND MAKE SURE THE DENOMINATOR IS IN FACTORED FORM.
SO FOR THIS FIRST FRACTION,
WE'LL WRITE THIS AS X / THE QUANTITY,
WHILE (6X - 36) WILL FACTOR,
THESE TWO TERMS SHARE A COMMON FACTOR OF 6,
SO WE CAN WRITE THIS AS 6 x THE QUANTITY (X -6),
AND THEN FOR -9 I'M GOING TO WRITE -9/1,
SO EVERYTHING IS IN FRACTION FORM.
AND THEN FOR THE RIGHT SIDE THE QUANTITY (X - 6) DOESN'T FACTOR,
SO WE'LL LEAVE IT AS 1/ THE QUANTITY (X - 6).
THE NEXT STEP IS TO IDENTIFY
WHAT WOULD BE THE LEAST COMMON DENOMINATOR.
WELL, THE LEAST COMMON DENOMINATOR
WOULD HAVE TO CONTAIN A FACTOR OF SIX AND A FACTOR OF (X - 6).
SO NOW WE'RE NOT GOING TO CONVERT TO A COMMON DENOMINATOR.
WE'RE ACTUALLY GOING TO MULTIPLY EACH FRACTION
BY 6 x THE QUANTITY (X - 6).
SO HERE WE'LL HAVE x 6 x THE QUANTITY (X - 6) ALL OVER 1,
THE SAME HERE,
AND THE SAME HERE.
BUT THERE'S ONE MORE IMPORTANT THING WE NEED TO DO
BEFORE WE SIMPLIFY
AND THEN FIND THESE PRODUCTS.
WE NEED TO LIST OUR EXCLUDED VALUES
WHERE THE VALUES OF X CAN NEVER BE
BECAUSE WE HAVE DIVISION BY 0.
NOTICE IF WE HAD A 6 HERE IN THE DENOMINATOR,
WE WOULD HAVE A DENOMINATOR OF 0 HERE AS WELL AS HERE.
SO WE MUST EXCLUDE 6 AS A POSSIBLE SOLUTION.
SO WE'LL GO AHEAD AND SAY X CANNOT EQUAL +6.
NOW THAT WE'VE LISTED THE EXCLUDED VALUES,
WE'LL GO AHEAD AND SIMPLIFY THEN FIND EACH OF THESE PRODUCTS.
LOOKING AT THIS FIRST PRODUCT, NOTICE HOW WE HAVE A 6/6.
THAT SIMPLIFIES TO 1.
WE ALSO HAVE A FACTOR OF (X - 6) OVER A FACTOR OF (X - 6).
NOTICE OUR DENOMINATOR IS NOW EQUAL TO 1,
AND OUR NUMERATOR IS JUST X.
SO NOW WE JUST HAVE X MINUS--
NOTHING SIMPLIFIES ON THIS PRODUCT,
BUT THAT'S OKAY BECAUSE OUR DENOMINATOR IS ALREADY 1.
SO WE HAVE 9 x 6 x THE QUANTITY (X - 6) EQUALS--
AND THEN ON THE RIGHT SIDE WE HAVE A FACTOR OF (X - 6)
OVER A FACTOR OF (X - 6).
THAT SIMPLIFIES TO 1.
AGAIN, NOTICE OUR DENOMINATOR IS EQUAL TO 1.
THE RIGHT SIDE OF THE EQUATION IS JUST 1 x 6 OR 6.
NOW WE'RE GOING TO GO AHEAD AND SOLVE THIS EQUATION,
SO WE HAVE X - THIS WOULD BE 54 x THE QUANTITY (X - 6) = 6.
LET'S GO AHEAD AND CLEAR THE PARENTHESES,
SO WE CAN THINK OF DISTRIBUTING A -54 HERE.
SO WE HAVE X, -54 x X, THAT'S GOING TO BE -54X.
-54 x -6 IS GOING TO BE +324, SO WE HAVE + 324 = 6.
WE'LL GO AHEAD AND COMBINE THE LIKE TERMS HERE.
WE HAVE 1X - 54X.
THAT'S -53X + 324 = 6.
WE'RE FINALLY DOWN TO A BASIC TWO-STEP EQUATION,
SO NOW WE'LL GO AHEAD AND SUBTRACT 324 FROM BOTH SIDES.
THIS IS 0.
WE HAVE -53X EQUALS THIS IS GOING TO BE -318,
AND NOW WE'LL DIVIDE BOTH SIDES BY -53.
SO THIS SIMPLIFIES TO X,
AND THEN -318 DIVIDED BY -53 IS EQUAL TO +6.
SO NOW IT MAY APPEAR THAT OUR SOLUTION WILL BE X = 6,
BUT REMEMBER +6 IS OUR EXCLUDED VALUE
THEREFORE THIS CANNOT BE A SOLUTION.
THEREFORE THIS EQUATION HAS NO SOLUTION.
THIS IS WHY IT'S SO IMPORTANT
THAT WHEN SOLVING A RATIONAL EQUATION
THAT WE TAKE THE TIME TO LIST THE EXCLUDED VALUES
AND THEN CHECK THEM AGAINST THE SOLUTIONS
THAT WE FIND ALGEBRAICALLY.
I REALIZE THAT IT'S A BIT OF A LETDOWN TO PERFORM ALL THIS WORK
AND THEN HAVE NO SOLUTION,
BUT SOMETIMES THIS IS THE OUTCOME.
I HOPE YOU FOUND THIS EXAMPLE HELPFUL.