- NOW WE'LL LOOK AT TWO EXAMPLES
OF SOLVING RATIONAL EQUATIONS.
BEFORE WE BEGIN, LET'S REVIEW THE STEPS
FOR SOLVING A RATIONAL EQUATION.
FIRST, WE NEED TO IDENTIFY THE VALUES OF X
THAT MAKE THE DENOMINATORS 0.
THESE X VALUES CANNOT BE USED AS SOLUTIONS TO THE EQUATION
SINCE DIVISION BY 0 IS UNDEFINED.
NEXT, WE'LL INDENTIFY THE COMMON DENOMINATOR,
HOPEFULLY THE LEAST COMMON DENOMINATOR
FOR ALL THE FRACTIONS IN THE EQUATION,
THEN WE'LL CLEAR THE FRACTIONS
BY MULTIPLYING BOTH SIDES OF THE EQUATION
BY THE COMMON DENOMINATOR
AND THEN WE'LL SOLVE FOR X
AND CHECK OUR SOLUTIONS USING SUBSTITUTION OR BY GRAPHING.
SO LOOKING AT OUR FIRST EQUATION
NOTICE HOW IF X = 4 WE'D HAVE A DENOMINATOR OF 0
AND THEREFORE, DIVISION BY 0.
SO WE SHOULD MAKE A NOTE THAT X CANNOT EQUAL +4.
IF WE DO GET +4 FOR A SOLUTION
WE WOULD HAVE TO EXCLUDE THAT VALUE.
FOR THE NEXT STEP IF IT'S HELPFUL WE CAN WRITE 3 AS 3/1
AND NOTICE HOW THE LEAST COMMON DENOMINATOR
WOULD BE THE QUANTITY X - 4.
SO NOW WE'LL MULTIPLY BOTH SIDES OF THE EQUATION BY X - 4
TO CLEAR THE FRACTION.
SO WE'D HAVE 3 x X - 4 = 23
OVER THE QUANTITY X - 4 x X - 4.
IF WE WANT TO WE COULD THIS X - 4 OVER 1.
SO ON THE LEFT SIDE, IF WE DISTRIBUTE WE'D HAVE 3X - 12
AND NOTICE ON THE RIGHT SIDE X - 4/X - 4 SIMPLIFIES TO 1
LEAVING US WITH JUST 23.
AND NOW, TO SOLVE FOR X WE WOULD ADD 12 TO BOTH SIDES.
SO WE HAVE 3X = 35,
LAST STEP TO SOLVE FOR X WE'D DIVIDE BOTH SIDES BY 3.
SO WE HAVE X = 35/3.
NOW LET'S CHECK THIS SOLUTION GRAPHICALLY.
TO DO THIS WE'LL ENTER Y SUB 1 = 3
ON THE GRAPHING CALCULATOR,
AND Y SUB 2 EQUALS THIS QUOTIENT
AND THEN WE'LL GRAPH THESE ON THE COORDINATE PLANE
AND LOOK FOR THE POINT OF INTERSECTION
TO VERIFY OUR SOLUTION.
BEFORE WE DO THIS THOUGH, THE CALCULATORS ONLY GOING
TO GIVE US A DECIMAL APPROXIMATION
WHEN WE FIND THE POINT OF INTERSECTION.
SO LET'S CONVERY 35/3 TO A DECIMAL.
THIS WOULD BE 11.6 REPEATING.
NOW WE'LL PRESS Y = ENTER Y1 = 3
AND Y2 IS GOING TO BE 23 DIVIDED BY THE QUANTITY X - 4.
WE DO NEED PARENTHESES AROUND THE ENTIRE DENOMINATOR
FOR Y SUB 2.
NOW IF WE USE THE STANDARD WINDOW BY PRESSING ZOOM 6
WE WON'T SEE THE POINT OF INTERSECTION
BECAUSE REMEMBER THE X COORDINATE
OF THE POINT OF INTERSECTION
IS SUPPOSE TO BE 11.6 REPEATING.
SO IN THIS CASE WE'LL HAVE TO INCREASE THE X MAXIMUM.
SO PRESS THE WINDOW KEY,
AND LET'S CHANGE THE X MAXIMUM TO 20
AND PRESS GRAPH AGAIN.
AND NOW TO FIND THE X COORDINATE
OF THIS POINT OF INTERSECTION
WE'LL PRESS SECOND TRACE
FOR THE CALCULATION MENU OPTION 5 FOR INTERSECTION
AND NOW FOR THE FIRST CURVE,
NOTICE HOW WE'RE ON THIS HORIZONTAL LINE WHICH IS GOOD.
SO WE'LL PRESS ENTER.
FOR THE SECOND CURVE,
WE WANT TO BE ON THIS PIECE OF THE CURVE HERE.
NOTICE HOW WE CANNOT SEE THE BLINKING CURSOR
BUT WE DO KNOW IT'S AT X = 5.
LET'S PRESS THE RIGHT ARROW HERE
AND MAKE SURE IT'S ON THE CORRECT PIECE OF THE CURVE
WHICH IT IS.
SO WE'LL PRESS ENTER,
WHEN IT SAYS GUESS WE CAN PRESS ENTER AGAIN.
NOTICE HOW THIS DOES VERIFY THAT OUR SOLUTION IS CORRECT.
X = 11.6 REPEATING, WHICH IS 35/3.
LOOKING AT OUR SECOND EXAMPLE,
NOTICE HOW X CAN'T EQUAL -4 OR X CAN'T EQUAL +3.
BECAUSE IN BOTH CASE WE WOULD HAVE DIVISION BY 0,
WHICH IS UNDEFINED.
ALSO NOTICE OUR LEAST COMMON DENOMINATOR
WOULD BE THE QUANTITY X + 4 x THE QUANTITY X - 3.
SO WE'LL MULTIPLY BOTH SIDES OF THE EQUATION
BY THIS PRODUCT.
SO WE'D HAVE 2
DIVIDED BY THE QUANTITY X + 4 x X + 4 x X - 3 = 7
DIVIDED BY THE QUANTITY X - 3 x THE SAME PRODUCT,
X + 4 x X - 3.
AGAIN IF IT'S HELPFUL WE CAN PUT THESE FACTORS OVER 1.
SO NOTICE ON THE LEFT SIDE, X + 4/X + 4 SIMPLIFIES TO 1.
AND ON THE RIGHT SIDE, X - 3/X - 3 SIMPLIFIES TO 1.
SO WE'RE LEFT WITH 2 x THE QUANTITY X - 3 = 7
x THE QUANTITY X + 4.
NOTICE HOW THE FRACTIONS ARE NOW CLEARED.
SO NOW WE'LL DISTRIBUTE AND SOLVE FOR X.
SO HERE WE'LL HAVE 2X - 6 AND HERE WE'LL HAVE 7X + 28.
LET'S GO AHEAD AND SUBTRACT 7X ON BOTH SIDES.
SO WE'D HAVE -5X - 6 = 28 ADD 6 TO BOTH SIDES.
WE HAVE -X 5X = 34, NOW WE DIVIDE BOTH SIDES BY -5.
AND WE HAVE X = -34/5.
AGAIN LET'S CHECK THIS GRAPHICALLY.
SO THIS WILL BE Y1 AND THIS WILL BE Y2.
LET'S BEGIN BY CONVERTING -34/5 TO A DECIMAL.
SO THE X COORDINATE OF THE POINT OF INTERSECTION
SHOULD BE -6.8.
SO NOW WE'LL PRESS Y = Y1 IS GOING TO BE 2
DIVIDED BY THE QUANTITY X + 4, ENTER.
LET'S GO AND CLEAR THIS
AND WE'LL ENTER 7 DIVIDED BY THE QUANTITY X - 3.
LET'S USE THE STANDARD WINDOW BY PRESSING ZOOM 6.
AND WE'RE LOOKING FOR THE INTERSECTION POINT HERE.
IT'S A LITTLE HARD TO SEE SO LET'S ADJUST THE WINDOW.
WE'LL PRESS WINDOW, LET'S DECREASE THE Y MINIMUM TO -20,
LET'S ALSO CHANGE THE Y VALUES FROM -5 TO +5.
AND LET'S TRY THIS BY PRESSING GRAPH.
SO WE'RE LOOKING FOR THE X COORDINATE
OF THIS POINT OF INTERSECTION HERE.
SE WE'LL PRESS SECOND TRACE OPTION 5.
NOTICE HOW WE ARE ON THE CORRECT PIECE
OF THE CURVE HERE,
SO I'LL PRESS ENTER.
NOTICE FOR THE SECOND CURVE WE'RE ON THE CORRECT PIECE.
WE DON'T WANT TO BE ON EITHER OF THESE PIECES HERE,
WE WANT TO MAKE SURE WE'RE ON BOTH OF THESE PIECES DOWN HERE
NEAR THE POINT OF INTERSECTION.
SO WE'LL PRESS ENTER AGAIN AND ENTER AGAIN.
AND NOTICE HOW ONCE AGAIN,
THIS DOES VERIFY OUR SOLUTION IS CORRECT.
I HOPE YOU FOUND THIS HELPFUL.