- TO SOLVE THE GIVEN RATIONAL EQUATION,
WE WANT TO MULTIPLY BOTH SIDES OF THE EQUATION BY AN EXPRESSION
THAT WILL ELIMINATE THE FRACTIONS.
THIS EXPRESSION WILL BE THE LEAST COMMON DENOMINATOR
OR LEAST COMMON MULTIPLE OF THE QUANTITY X + 6,
THE QUANTITY X - 3, AND THE QUANTITY X SQUARED + 3X - 18.
SO TO DETERMINE THE LEAST COMMON DENOMINATOR,
WE DO WANT TO MAKE SURE THE DENOMINATORS
ARE IN FACTORED FORM
SO WE'LL START BY FACTORING THIS.
THE FACTORS OF X SQUARED ARE X AND X.
THE FACTORS OF -18 THAT ADD TO 3 ARE +6 AND -3.
THIS IS ACTUALLY GOOD NEWS,
BECAUSE NOTICE HOW THESE 2 FACTORS DO OCCUR
IN THE OTHER 2 DENOMINATORS.
NOW THAT ARE DENOMINATORS ARE FACTORED,
WE SHOULD BE ABLE TO RECOGNIZE
THAT THE LEAST COMMON DENOMINATOR
WOULD BE THE QUANTITY X + 6 x THE QUANTITY X - 3.
SO NOW INSTEAD OF OBTAINING A COMMON DENOMINATOR,
WE'RE GOING TO MULTIPLY EACH TERM IN THE EQUATION
OR BOTH SIDES OF THE EQUATION BY THIS PRODUCT.
SO NOW, WE'RE GOING TO GO AHEAD
AND REWRITE THE ORIGINAL EQUATION
LEAVING PLENTY OF SPACE HORIZONTALLY.
AND NOW, WE'LL MULTIPLY EACH FRACTION BY THE LCD.
SO WE'LL HAVE x THE QUANTITY X + 6 x THE QUANTITY X - 3.
WE'LL PUT THIS OVER 1.
IT'S NOT REQUIRED TO DO THIS,
BUT IT WILL MAKE IT EASIER TO SEE
WHAT'S GOING TO SIMPLIFY OUT.
AND NOW, WE'LL DO THE SAME TO THE OTHER 2 FRACTIONS.
AND NOW, WE'LL SIMPLIFY.
NOTICE HERE WE HAVE A FACTOR OF X + 6/X + 6
WHICH SIMPLIFIES TO 1,
HERE WE HAVE A FACTOR OF X - 3/X - 3,
AND HERE WE HAVE A FACTOR OF X + 6/X + 6 AND X - 3/X - 3.
NOW, WE'LL DETERMINE THESE PRODUCTS
AND THEN SOLVE THE RESULTING EQUATION.
NOTICE HERE WE HAVE THE QUANTITY X - 3 x THE QUANTITY X - 3,
PLUS HERE WAS HAVE THE QUANTITY X - 2 x THE QUANTITY X + 6.
AND ON THE RIGHT SIDE, WE JUST HAVE X SQUARED.
NOW, WE'LL CLEAR THE PARENTHESES.
SO WE'LL HAVE X SQUARED - 3X - 3X, THAT'S - 6X
+ 9 + X SQUARED + 6X - 2X, THAT'S +4X - 12 = X SQUARED.
NOW, WE'LL COMBINE LIKE TERMS ON THE LEFT.
WE HAVE 1X SQUARED + 1X SQUARED, THAT'S 2X SQUARED,
-6X + 4X, THAT'S -2X,
AND THEN 9 - 12 IS -3, SO WE HAVE -3 = X SQUARED.
NOW, WE'LL SET IT EQUAL TO 0 BY SUBTRACTING X SQUARED
ON BOTH SIDES.
SO THEN NOTICE THE RIGHT SIDE WOULD BE 0,
AND WE HAVE X SQUARED - 2X - 3.
LET'S GO AHEAD AND MOVE THIS OVER HERE AND FACTOR THIS.
THE FACTORS OF X SQUARED ARE X AND X.
THE FACTORS OF -3 THAT HAVE THE -2 WOULD BE -3 AND +1.
THIS PRODUCT WILL BE EQUAL TO 0 WHEN X IS EQUAL TO 3
BECAUSE THAT WOULD MAKE THIS FACTOR EQUAL TO 0,
OR WHEN X IS EQUAL TO -1
BECAUSE -1 WOULD MAKE THIS FACTOR EQUAL TO 0.
NOW BEFORE WE SAY THESE ARE THE SOLUTIONS,
WE DO NEED TO GO BACK UP TO THE EQUATION JUST FOR A MOMENT.
WE KNOW DIVISION BY 0 IS UNDEFINED,
SO WE MUST EXCLUDE THE VALUES OF X
THAT WOULD MAKE THESE DENOMINATORS EQUAL TO 0.
WELL, NOTICE IF X IS EQUAL TO -6,
IT WOULD MAKE THIS DENOMINATOR EQUAL TO 0 AS WELL AS THIS ONE,
SO WE MUST EXCLUDE -6.
BUT NOTICE ALSO,
+3 WOULD MAKE THIS DENOMINATOR AND THIS DENOMINATOR EQUAL TO 0,
SO X ALSO CANNOT EQUAL +3.
AND THIS DOES CONFLICT WITH ONE OF OUR SOLUTIONS HERE
WHERE WE SAY X = 3.
X CANNOT = 3 HERE,
BECAUSE IT WOULD MAKE THE DENOMINATOR EQUAL TO 0.
AND THEREFORE WE ONLY HAVE ONE SOLUTION, WE WOULD HAVE X = -1.
SO IT IS IMPORTANT THAT WE ALWAYS MAKE SURE
THAT WE EXCLUDE THE VALUES
THAT WOULD MAKE THE DENOMINATORS EQUAL TO 0
BEFORE WE STATE OUR FINAL SOLUTION.
OKAY, I HOPE YOU FOUND THIS HELPFUL.