- 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.