- WELCOME TO ABSOLUTE VALUE EQUATIONS.
THE GOALS OF THIS VIDEO ARE TO SOLVE ABSOLUTE VALUE EQUATIONS
ALGEBRAICALLY
AND SOLVE THEM GRAPHICALLY.
LET'S START OFF BY REVIEWING THE DEFINITION OF ABSOLUTE VALUE.
A NUMBER'S ABSOLUTE VALUE IS ITS DISTANCE FROM ZERO
ON THE NUMBER LINE
AND AS WE KNOW A DISTANCE IS ALWAYS GOING TO BE POSITIVE.
SO FOR EXAMPLE, THE ABSOLUTE VALUE OF +3 IS EQUAL TO 3
SINCE THIS NUMBER IS 3 UNITS FROM ZERO.
WELL THE ABSOLUTE VALUE OF -3 IS ALSO EQUAL TO 3
BECAUSE -3 IS ALSO 3 UNITS FROM ZERO.
SO WHEN WE TRANSITION TO AN ABSOLUTE VALUE EQUATION,
WE CAN THINK OF WHAT NUMBERS HAVE AN ABSOLUTE VALUE
THAT IS EQUAL TO +2?
OR WHAT NUMBERS ON THE NUMBER LINE ARE TWO UNITS FROM ZERO?
AND WE SHOULD BE ABLE TO NOTICE NOW
THAT THERE ARE GOING TO BE TWO SOLUTIONS TO THIS EQUATION
BECAUSE +2 IS TWO UNITS FROM ZERO AND SO IS -2.
SO THE SOLUTION TO THIS EQUATION WOULD BE X = +2 AND X = -2.
SO LET'S TALK ABOUT THE ABSOLUTE VALUE PRINCIPLE FOR EQUATIONS
FOR ANY POSITIVE NUMBER C AND ANY ALGEBRAIC EXPRESSION X,
THE SOLUTIONS OF THE ABSOLUTE VALUE OF X EQUALS THE C,
A POSITIVE VALUE
ARE THOSE NUMBERS THAT SATISFY THE EQUATION X = -C OR X = +C.
NOW REMEMBER WE'RE USING X HERE
BUT THIS COULD BE ANY EXPRESSION INVOLVING X.
THERE ARE TWO OTHER CASES.
IF WE HAVE THE ABSOLUTE VALUE OF X = 0,
THIS IS EQUIVALENT TO THE EQUATION X = 0
BECAUSE KNOW +0 AND -0 RESULTS IN 0.
AND THE LAST CASE IS WHEN WE ADD THE ABSOLUTE VALUE OF X = -C
AND THIS HAS NO SOLUTION BECAUSE WE SHOULD KNOW NOW
THAT THE ABSOLUTE VALUE MEANS A DISTANCE
AND DISTANCE CAN NEVER BE ZERO.
SO HERE ARE THE STEPS WE'RE GOING TO TAKE TO SOLVE
ABSOLUTE VALUE OF EQUATIONS.
STEP ONE, STEP ONE WE MUST ISOLATE THE ABSOLUTE VALUE.
STEP TWO WE WILL SET UP AND SOLVE TWO EQUATIONS
BASED UPON THE ABSOLUTE VALUE PRINCIPLE,
WHERE WE HAVE THE ABSOLUTE VALUE OF X = +C.
SO WE'LL TAKE THE EXPRESSION INSIDE THE ABSOLUTE VALUE SYMBOL
SET IT EQUAL TO -C AND +C AND SOLVE.
AND THEN LASTLY WE'LL CHECK OUR ANSWERS.
LET'S GO AHEAD AND GIVE IT A TRY.
THE ABSOLUTE VALUE OF X IS EQUAL TO 6.
OUR ABSOLUTE VALUE IS ALREADY ISOLATED.
SO OUR TWO EQUATIONS WILL BE X IS EQUAL TO +6
AND X IS EQUAL TO -6.
AND JUST TO CHECK THIS, THE ABSOLUTE VALUE OF 6 IS +6
AND THE ABSOLUTE VALUE OF -6 IS EQUAL TO +6, THAT CHECKS.
OKAY RIGHT AWAY ON NUMBER TWO THEY'RE TESTING US.
WE ADD THE ABSOLUTE VALUE OF X + 1 = -2.
WELL WE KNOW AN ABSOLUTE VALUE CAN NEVER EQUAL
A NEGATIVE NUMBER,
THEREFORE THIS EQUATION WILL HAVE NO SOLUTION.
HOWEVER, IF YOU DON'T RECOGNIZE THAT RIGHT AWAY
AND YOU DO TRY TO SET UP YOUR TWO EQUATIONS,
AS X + 1 = -2 AND X + 1 = +2,
WE CAN STILL SOLVE THESE, SUBTRACT ONE ON BOTH SIDES.
AGAIN SUBTRACT ONE.
NOW WE MAY THINK THESE WOULD BE SOLUTIONS
BUT IF WE ACTUALLY GO BACK AND CHECK THEM USING X = -3,
WE'D HAVE THE ABSOLUTE VALUE OF -2 = -2 WHICH IS FALSE.
AND THEN WHEN X = 1, AGAIN IT'S FALSE
BECAUSE WE KNOW IT CAN NEVER BE NEGATIVE.
SO IF WE RECOGNIZE THIS FROM THE VERY BEGINNING
THAT WE'RE GOING TO HAVE NO SOLUTION,
WE CAN STOP AND MOVE ON TO THE NEXT PROBLEM.
IF YOU DON'T RECOGNIZE THAT AND SOLVE,
WE'VE JUST ILLUSTRATED WHY IT'S IMPORTANT
THAT YOU DO HAVE TO CHECK YOUR WORK.
LET'S LOOK AT A COUPLE MORE AND THEN WE'LL TAKE A LOOK AT
HOW WE CAN SOLVE THESE GRAPHICALLY.
SO IT IS IMPORTANT
THAT WE ISOLATE THE ABSOLUTE VALUE FIRST.
SO IN THIS PROBLEM WE'LL ADD 3 TO BOTH SIDES,
THAT WOULD RESULT IN THE ABSOLUTE VALUE OF 2X + 3 = 9.
THIS IS THE FORM OF THE EQUATION THAT WE USE
TO SET UP AND SOLVE OUR TWO EQUATIONS.
IF WE TRY TO USE IT IN THIS FORM, AND USE A +6 OR -6,
WE WILL NOT GET THE CORRECT ANSWERS.
AND THAT'S A VERY COMMON ERROR THAT I SEE.
SO OUR FIRST EQUATION WOULD BE 2X + 3 CAN EQUAL 9
AND 2X + 3 CAN ALSO EQUAL -9.
AND HOPEFULLY THIS MAKES SENSE
AS TO WHY WE'RE SETTING THESE TWO EQUATIONS UP.
IT COMES FROM THE FACT THAT THE ABSOLUTE VALUE OF 9 = 9
AND SO IS THE ABSOLUTE VALUE OF -9.
SO THIS 9 IS REALLY THIS 9 HERE
AND THIS -9 IS REALLY THIS -9 HERE.
LET'S GO AHEAD AND SOLVE THIS.
SUBTRACT 3 ON BOTH SIDES, DIVIDE BY 2, SAME PROCEDURES HERE,
BUT NOW WHEN WE SUBTRACT THREE ON BOTH SIDES
WE'LL HAVE -12 ON THE RIGHT, DIVIDE BY 2, X = -6.
SO OUR SOLUTIONS ARE X = 3 AND X = -6.
YOU CAN SEE AS SOON AS YOU SET UP YOUR TWO EQUATIONS,
IT'S PRETTY STRAIGHTFORWARD.
IT'S JUST MAKING SURE YOU FOLLOW THE CORRECT TECHNIQUES
TO SET UP THESE TWO LINEAR EQUATIONS.
NUMBER FOUR, HERE IS THE ABSOLUTE VALUE PIECE
THAT WE MUST ISOLATE.
SO FIRST WE'LL SUBTRACT 4 ON BOTH SIDES.
12 - 4 WOULD GIVE US 8.
THIS IS CONNECTED BY MULTIPLICATION
SO WE'LL DIVIDE BOTH SIDES BY +2.
SO WE HAVE THE ABSOLUTE VALUE OF 4X - 2 = +4.
YOU WILL HAVE 2 LINEAR EQUATIONS TO SOLVE.
THE FIRST ONE WILL BE 4X -2 = +4 OR 4X - 2 COULD EQUAL -4.
SO HERE WE'LL ADD 2 TO BOTH SIDES, 4X = 6 DIVIDED BY 4,
X = 3/2.
SAME STEPS, ADD 2 TO BOTH SIDES -4 + 2 WOULD BE -2.
AGAIN DIVIDE BY +4, WE HAVE X = -1/2.
AS EXPECTED WE HAVE TWO SOLUTIONS.
SO YOU CAN SEE THE PATTERN HERE,
WE CAN EXPECT TWO SOLUTIONS TO MOST ABSOLUTE VALUE EQUATIONS
UNLESS THE ABSOLUTE VALUE IS EQUAL TO A NEGATIVE NUMBER
WHERE WE HAVE NO SOLUTIONS,
OR WHERE IT'S EQUAL TO ZERO AND WE WOULD HAVE ONE SOLUTION.
LET'S TAKE A LOOK AT HOW WE CAN SOLVE
ABSOLUTE VALUE EQUATIONS GRAPHICALLY.
STEP ONE, GRAPH THE LEFT SIDE OF THE EQUATION, IN Y1.
GRAPH THE RIGHT SIDE OF THE EQUATION IN Y2,
DETERMINE THE POINTS OF INTERSECTION
AND THE X COORDINATES OF THE POINTS OF INTERSECTION
ARE THE SOLUTIONS.
LET'S GO AHEAD AND GIVE IT A TRY.
SO WE'RE GOING TO GRAPH THE ABSOLUTE VALUE OF 2X + 1 IN Y1
AND WE'LL GRAPH 5 IN Y2.
PRESS Y =.
IF YOU HAVE ANY OLD EQUATIONS IN THERE, CLEAR THOSE OUT.
TO ACCESS THE ABSOLUTE VALUE, PRESS MATH, RIGHT ARROW ONCE,
A.B.S. STANDS FOR ABSOLUTE VALUE, PRESS ENTER.
TYPE IN X + 2.
TYPE IN X + 2 CLOSE THE PARENTHESES.
THERE'S Y1, Y2 = 5.
TO MAKE SURE WE HAVE THE STANDARD WINDOW,
LET'S HIT ZOOM 6.
WE'LL SEE TWO POINTS OF INTERSECTION.
THESE X COORDINATES OF THESE POINTS REPRESENT THE SOLUTIONS.
IF WE PRESS SECOND TRACE, OPTION 5,
THE CALCULATOR CAN ONLY FIND ONE POINT OF INTERSECTION AT A TIME.
WE WANT TO MOVE THE CURSOR CLOSE TO THE POINT OF INTERSECTION
WE WANT TO FIND FIRST.
LET'S FIND THE ONE ON THE RIGHT FIRST.
PRESS ENTER THREE TIMES, THEN WE CAN SEE ONE SOLUTION IS X = 3.
LET'S RECORD THIS.
NOW WE'LL GO BACK AND FIND THE SECOND SOLUTION,
PRESS SECOND TRACE OPTION 5,
NOW MOVE THE CURSOR CLOSE TO THE LEFT POINT OF INTERSECTION,
PRESS ENTER THREE TIMES, AND WE HAVE X = -7.
THAT WAS PRETTY QUICK AND PAINLESS
BUT LET'S GO AHEAD AND CHECK OUR WORK.
FIRST X = 3 WHICH SATISFIES THE EQUATION.
NEXT X = -7 AND THE ABSOLUTE VALUE OF -5 = 5
SO BOTH SOLUTIONS CHECK.
LET'S TRY ONE MORE.
NOW IN THIS ONE AGAIN WE DO NOT HAVE TO
ISOLATE THE ABSOLUTE VALUE FOR SOLVING GRAPHICALLY.
THIS WILL BE Y1 AND THIS WILL BE Y2.
REMEMBER TO ACCESS THE ABSOLUTE VALUE WE HIT MATH,
RIGHT ARROW ONCE AND THEN ENTER,
CLOSE THE SET OF PARENTHESES.
THERE'S Y1, Y2 WILL BE 11
NOW THIS WILL BE AN ISSUE
BECAUSE REMEMBER THE STANDARD WINDOW ONLY GOES UP TO 10,
SO WE'LL HAVE TO CHANGE OUR Y MAXIMUM.
NOTICE IF WE PRESS GRAPH, WE DO NOT SEE THE HORIZONTAL LINE.
SO IF WE PRESS WINDOW, AND CHANGE OUR Y MAXIMUM
SO THAT IT INCLUDES 11,
I'LL CHANGE IT TO 15 AND NOW PRESS GRAPH.
WE NOW CAN SEE THE POINTS OF INTERSECTION,
SO AGAIN PRESS SECOND TRACE, OPTION 5.
I'LL FIND THE NEGATIVE SOLUTION FIRST, ENTER THREE TIMES, X = -2
AND THE SECOND SOLUTION X = 3.
I HOPE YOU FOUND THIS VIDEO HELPFUL.