WELCOME TO OUR VIDEO
ON SOLVING RATIONAL EQUATIONS. LET'S GO AHEAD
AND GET STARTED. HERE ARE THE GUIDELINES FOR
SOLVING RATIONAL EQUATIONS. STEP ONE, WE WILL FACTOR
ALL THE DENOMINATORS AND STEP TWO,
WE'LL DETERMINE THE LCD AND THEN MULTIPLY EACH SIDE
OF THE EQUATION BY THE LCD OR LEAST COMMON DENOMINATOR. THEN, WE'LL SOLVE
THE RESULTING EQUATION AND THEN, WE'LL CHECK
OUR SOLUTIONS. WHILE WE CHECK OUR SOLUTIONS,
WE NEED TO BE AWARE THAT IF WHAT APPEARS TO BE
A SOLUTION MAKES A DENOMINATOR
EQUAL TO ZERO, WE MUST EXCLUDE THAT VALUE. AND THEN LASTLY, WE CAN CHECK OUR SOLUTIONS
ON THE GRAPHING CALCULATOR. IF WE TYPE THE LEFT SIDE
OF THE EQUATION IN Y1 AND THE RIGHT SIDE
OF THE EQUATION IN Y2, WE CAN CHECK OUR SOLUTIONS
BY FINDING THE POINT OF POINTS OF INTERSECTION
OF THE TWO GRAPHS. LET'S GO AHEAD
AND TAKE A LOOK AT AN EXAMPLE. NOW, WE CAN PROBABLY DETERMINE
THE LCD OR LEAST COMMON DENOMINATOR
OF THIS PROBLEM WITHOUT FACTORING
THE DENOMINATORS. THE LCD IN THIS CASE
WOULD BE 6T. THAT MEANS,
WE'RE GOING TO MULTIPLY BOTH SIDES OF THE EQUATION
BY 6T OR 6T OVER 1. BUT WHAT I'M GOING TO DO
IS I'M GOING TO MULTIPLY EVERY SINGLE TERM BY 6T OVER 1 BECAUSE IT MAKES SIMPLIFYING
THOSE PRODUCTS A LOT EASIER. THIS IS WHAT I MEAN BY THAT. I'LL TAKE EACH OF THESE TERMS
AND MULTIPLY BY 6T OVER 1. NOW, WE'LL SIMPLIFY
EACH OF THESE PRODUCTS, THIS WOULD CHANGE TO A 1,
THIS WOULD CHANGE TO A 2 AND HERE, THE 6S SIMPLY OUT AND HERE THE T
IS SIMPLIFIED OUT. LET'S SEE WHAT REMAINS. HERE, WE HAVE 2 TIMES 2
TIMES T ALL OVER 1 THAT'S 4T. HERE, WE HAVE 5 TIMES T,
SO WE HAVE MINUS 5T AND HERE, WE HAVE 1 TIMES 6
ALL OVER 1, WHICH IS JUST 6. SO, WE HAVE CREATED
AN EQUATION THAT IS MUCH EASIER TO SOLVE THAN THE ORIGINAL
RATIONAL EQUATION. SO, HERE WE CAN COMBINE
OUR LIKE TERMS. THIS WOULD BE NEGATIVE T
EQUALS 6. YOU COULD THINK OF THIS
AS NEGATIVE 1T DIVIDED BY NEGATIVE 1. SO, WE HAVE T EQUALS
NEGATIVE 6 LET'S GO AHEAD
AND TRY ANOTHER. EVEN THOUGH
THIS IS A PROPORTION, I'M GOING TO GO AHEAD
AND FOLLOW THE RULES OUTLINED IN THIS VIDEO. WE CAN SEE THE LCD
WOULD BE THE PRODUCT OF THESE TWO DENOMINATORS. SO, WE'LL MULTIPLY BOTH SIDES
OF THE EQUATION BY THIS PRODUCT OVER 1. SO NOTICE IN RED, WE'VE MULTIPLIED
BY SIDES OF THE EQUATION BY X MINUS 6 TIMES X PLUS 2
ALL OVER 1. NOW, LET'S GO AHEAD
AND SIMPLIFY. NOTICE, WE HAVE A COMMON
FACTOR OF X MINUS 6 HERE. SO, WE'RE LEFT WITH X PLUS 2
TIMES X PLUS 2 MUST EQUAL. OVER HERE,
WE HAVE AN X PLUS 2 ON TOP, X PLUS 2 ON THE BOTTOM. SO, WE'RE LEFT WITH X MINUS 1
TIMES X MINUS 6. SO, THE FRACTIONS ARE GONE, BUT NOW WE HAVE TO MULTIPLY
THIS OUT AND SOLVE FOR X. SO, WE'RE GOING TO HAVE
TO FOIL BOTH SIDES HERE. WE'LL HAVE X SQUARED
PLUS 2X, PLUS 2X, THAT'S PLUS 4X, PLUS 4 EQUALS. HERE, WE HAVE X SQUARED
MINUS 6X MINUS 1X, THAT'S MINUS 7X, PLUS 6. NOW, WE'RE LUCKY HERE BECAUSE IF WE SUBTRACT
X SQUARED ON BOTH SIDES, THE X SQUARED TERM GOES OUT. SO, WE DO HAVE
A LINEAR EQUATION LEFT. LET'S GO AHEAD
AND ADD 7X ON BOTH SIDES. WE WOULD HAVE
11X PLUS 4 EQUALS 6 AND NOW WE'LL SUBTRACT 4
ON BOTH SIDES. I'M RUNNING OUT OF ROOM HERE. LET'S GO AHEAD
AND BRING THIS UP HERE. WE WOULD HAVE 11X EQUALS 2,
DIVIDE BY 11, X EQUALS 2/11. REMEMBER THE GUIDELINES
DID SAY CHECK OUR SOLUTION, BUT REALLY,
WHAT WE'RE CHECKING FOR IS X CAN NEVER EQUAL 6
OF NEGATIVE 2 BECAUSE IT WOULD MAKE THESE
DENOMINATORS EQUAL TO ZERO AND THEREFORE, UNDEFINED. SO, X EQUALS 2/11
WE'LL ASSUME. WE'LL CHECK. HOWEVER, LET'S GO AHEAD
AND CHECK THIS ONE ON THE GRAPHING CALCULATOR. FIRST, LET'S CHECK 2 DIVIDED
BY 11 IN DECIMAL FORM IS 0.18 REPEATING. LET'S GO TO Y EQUALS. LET'S TYPE THE LEFT SIDE IN Y1
AND THE RIGHT SIDE IN Y2. LET'S GO AHEAD
AND PRESS GRAPH. AGAIN, WE'RE LOOKING
FOR THE X COORDINATE OF THE POINT OF INTERSECTION. AND IT LOOKS LIKE IT'S
SOMEWHERE RIGHT IN HERE. LET'S GO AHEAD
AND CALCULATE IT. SECOND TRACE, OPTION FIVE,
PRESS ENTER, ENTER, ENTER AND THIS DOES VERIFY
OUR SOLUTION AS 2/11. LET'S GO AHEAD
AND TRY ANOTHER ONE. OKAY, ON THIS PROBLEM, WE ARE GOING TO HAVE TO FACTOR
THE DENOMINATORS TO DETERMINE THE LCD. LET'S GO AHEAD AND FACTOR
THIS DENOMINATOR HERE. THE FACTORS OF NEGATIVE 6
THAT ADD TO POSITIVE 1. THAT WOULD BE POSITIVE 3
AND NEGATIVE 2 ARE THE SAME FACTORS
WE SEE IN THIS DENOMINATORS. SO, THE LCD WOULD BE X PLUS 3
TIMES X MINUS 2. SO, WE NEED TO MULTIPLY
EACH TERM IN THIS EQUATION BY X PLUS 3 TIMES X MINUS 2,
ALL OVER 1 LET'S GO AHEAD AND DO THAT. SO, NOTICE IN BLUE, I HAVE THE
ORIGINAL RATIONAL EXPRESSIONS. I DID WRITE THE RIGHT SIDE
IN FACTORED FORM AND THEN IN RED, I'VE MULTIPLIED EVERY TERM
BY THE LCD OVER 1. LET'S GO AHEAD AND SIMPLIFY
BEFORE WE FIND THIS PRODUCTS. NOTICE, WE HAVE AN X PLUS 3
AND AN X PLUS 3. SO, WE'RE LEFT
WITH X TIMES X MINUS 2. ON THE NEXT PRODUCT,
WE HAVE X MINUS 2 ON TOP, X MINUS 2 ON THE BOTTOM. SO, WE'RE LEFT WITH MINUS X
TIMES X PLUS 3 EQUALS. AND ON THE RIGHT SIDE,
NOTICE WE HAVE X PLUS 3, X PLUS 3, X MINUS 2, X MINUS 2 SO, WE'RE LEFT WITH JUST 10
ON THE RIGHT SIDE. LET'S GO AHEAD AND SOLVE
THIS RESULTING EQUATION. SO, WE'LL DISTRIBUTE
AND THEN SOLVE FOR X. SO, I HAVE X SQUARED,
MINUS 2 X, MINUS X SQUARED, MINUS 3X EQUALS 10. NOTICE THE X SQUARED TERM
SIMPLIFY OUT. WE'RE LEFT WITH NEGATIVE 5X
EQUALS 10. DIVIDING BY NEGATIVE 5,
X EQUALS NEGATIVE 2. AGAIN, THIS WILL BE
A SOLUTION. IF THE SOLUTION CAME OUT TO BE
A POSITIVE 2 OR A NEGATIVE 3, WE WOULD HAVE TO EXCLUDE IT. LET'S GO AHEAD
AND TRY ONE MORE. AGAIN, WE'RE GOING TO HAVE
TO FACTOR THIS DENOMINATOR TO DETERMINE LCD. THE FACTORS OF NEGATIVE 2
THAT ADD TO A POSITIVE 1, THAT WOULD BE PLUS 2
AND MINUS 1. SO, LOOKING AT THE FACTORS
OF THE DENOMINATORS, WE CAN SEE THE LCD WOULD BE Y
PLUS 2 TIMES Y MINUS 1. SO, WE'LL MULTIPLY
EACH RATIONAL EXPRESSION BY THE LCD OVER 1. AGAIN, IN BLUE, WE HAVE THE
ORIGINAL RATIONAL EXPRESSIONS AND IN RED,
WE HAVE THE LCD OVER 1. NOW, WE'LL FIND EACH PRODUCT,
BUT WE'LL SIMPLIFY FIRST. SO, WE HAVE Y PLUS 2
OVER Y PLUS 2. SO, WE'RE LEFT WITH 4Y TIMES Y
MINUS 1, MINUS. HERE, WE HAVE A Y MINUS 1
AND A Y MINUS 1. SO, WE HAVE MINUS 3Y
TIMES THE QUANTITY Y PLUS 2. ON THE RIGHT SIDE, WE HAVE A Y PLUS 2, Y PLUS 2,
Y MINUS 2, Y MINUS 1. SO, WE'RE LEFT WITH Y SQUARED
MINUS 8Y, MINUS 4. WE NEED TO CLEAR
THESE PARENTHESES AND THEN, SOLVE FOR Y. SO, 4Y SQUARED MINUS 4Y,
MINUS 3Y SQUARED, MINUS 6Y EQUALS Y SQUARED,
MINUS 8Y, MINUS 4. COMBINING LIKE TERMS,
WE HAVE Y SQUARED, MINUS 10Y EQUALS Y SQUARED
MINUS 8Y, MINUS 4. LET'S GO AHEAD TAKE THIS
OVER TO THE NEXT SCREEN AND FINISH SOLVING THIS. AND YOU'LL NOTICE, IF WE SUBTRACT Y SQUARED
ON BOTH SIDES, THE Y SQUARED TERMS
ARE ELIMINATED. SO, WE'RE LEFT WITH NEGATIVE
10Y EQUALS NEGATIVE 8Y, MINUS 4. LET'S GO AHEAD
AND ADD 8Y TO BOTH SIDES. WE HAVE NEGATIVE 2Y
EQUALS NEGATIVE 4, DIVIDED BY NEGATIVE 2. IT LOOKS LIKE WE HAVE Y
EQUALS POSITIVE 2. NOW, AGAIN BEFORE WE CAN
CONCLUDE THIS IS A SOLUTION, WE NEED TO GO BACK AND CHECK TO MAKE SURE IT DOESN'T MAKE
ONE OF OUR DENOMINATORS IN THE ORIGINAL EQUATION ZERO. AND NEGATIVE 2 WOULD NOT WORK, BUT POSITIVE 2
WILL BE JUST FINE. OKAY. SO, AS YOU SEE,
THESE CAN BE FAIRLY INVOLVED. I HOPE YOU FOUND THIS VIDEO
HELPFUL.