# Absolute Value Equations

– 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
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

### 6 thoughts on “Absolute Value Equations”

• September 28, 2011 at 7:52 pm

Thank you again!

• November 10, 2012 at 10:44 pm

Me too!

• October 16, 2013 at 3:40 am

I approve of this video.

• January 28, 2016 at 8:35 pm

This video was really helpful !