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How To Solve Absolute Value Equations
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Feb 19, 2025
In this lecture, the students and I are going to learn how to solve simple and complex absolute value equations. But first, we'll learn how to evaluate absolute value expressions Chapters 00:00 Introduction 00:21 How To evaluate absolute value expressions 08:39 How To solve absolute value equations ( real life example ) 16:47 Example 2
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0:01
all right so today we are talking about
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solving equations
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with absolute value all right again now
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things are getting a little bit more
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complex all right but before we do it
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before we get to the equations we need
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to learn how to evaluate an absolute
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value expression that's just the basics
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right you guys know about absolute value
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yes you don't I'm sure you've done CU I
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did it in prealgebra and algebra one
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right so let me give you a quick example
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so if you have absolute value of a it
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can take on two values
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right a and
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normal fantastic this good I like that
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fantastic love that all right so abute
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value of a could be negative a or a why
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is the two
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values because we don't know what a is
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right if it's if a is positive the abute
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value of positive value is what
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a positive value right assume that a was
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two what would be absolute value of
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two two right and if a was -2 what's
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abute value of -2 two right who say -2 I
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heard -2 it's not it's positive to right
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CU absolute value is is always what
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positive right so basically this is the
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key here we don't know a but we know
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that if a is negative the absolute value
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of a will be a double negative which
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gives you positive and if a is positive
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you just keep it as it is right so now
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we're going to learn how to just
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evaluate it so I'm just going to plug
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in so I want to find what this is here
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negative absolute value of 2x + 5 if x
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is -7 maybe what do I do
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here I want to evaluate this expression
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like we know this is an expression right
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because it has what variables end
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numbers so I want to evaluate it I want
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to find what it is I want to solve it so
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what am I going to do I know that X is
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-7 so what am I doing here yes
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sir you are plugging -7 for X right so
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we get -2 * -7 + 5 right so now before
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you solve this you have to solve the
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inside first it's just like the pmas
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right we solve what's in here and then
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we find the absolute value does that
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make
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sense so what's -2 *
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-7 14 14 right + 5 so now I have
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negative absolute value of 19 what would
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that give me 19 are you sure it's
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19 19a 19 because absolute value of 19
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is 19 and the negative where inside or
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outside outside so therefore it's
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negative you get the
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yeah it started started oh it's positive
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no no no that's not a I was that was not
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like a mistake that's just
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Mis not miscalculation I just thought it
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was negative to so that don't count
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that's a mistake no that don't count you
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would take off on a test if I did that
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no I wouldn't I I'll tell you just pay
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attention got a different answer yeah
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all right that's right all right so -4 +
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5 which is POS 9 all right so it's
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negative value of 9 and that gives you
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what 9 thank you right 9 all right so
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this is just the basics so now I want
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you to work on this on your own you have
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4x +
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3 - 3 and then 12 and then I want you to
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finding when X is
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-2 okay so just do that on your own real
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quick and then you see what you
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got can I have a
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calculator yes you may anybody El need a
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calculator can I have a mar you want a
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car can I have a
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mar what you need is that 14 or right
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here yeah oh this is absolute value of
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4X it's absolute value y you done wow
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Speedy Gonzalez
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fraction what
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you anybody
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done not yet so yeah all
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[Music]
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[Music]
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right should
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be
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is5 all right so that's 10us
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can turn into FR
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[Music]
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right then I have to find the same
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denominator
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what
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two yeah right 10 - 7
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[Music]
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[Music]
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what bottom
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all right so let's do it over here right
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so I'm going first X -2 so we got 4 *
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-2 + 3 - 3 and 12
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right 4 * -2
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is8 and then 8 + 3 is5 so we get
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absolute value
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of5 3 and 12 right absolute value of 5
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is
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5 now I can turn this into a improper
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fraction right that gives me 72
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everyone know I got the 72 all
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right 5- 72 now we have to find the
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LCD two here times this by two so we get
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10 on your s over two it should be three
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half some people 1.5 like I say this is
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fine 1.5 is fine but get into the habit
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of getting fractions I do not want to
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see decimals unless we are starting with
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decimal right get into the of getting
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fractions because we are doing alra two
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now so we need to start using regular
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like decimal push yes sir I did it just
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5 half but I did it just off just sense
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what you got and half yeah that's fine
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yeah but I didn't do like the 10 no you
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don't have to do do it like this if you
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know how to do it like mentally that's
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fine I was just showing the details
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right anybody has a problem with this
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does everybody understand this we all
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good are we good all right now the next
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thing is now we're going to talk about
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absolute value equations okay so example
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one a standard adult tennis bracket has
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about 100 in square head right so if you
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have a tennis rocket it looks like this
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right all the bre or whatever I don't
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play that much tennis but this is the
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head so the standard head is about 100
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in square that's what this is telling us
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Square in right area right yeah area
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plus orus 20 in squ so that means it
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varies it could be either more
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than or less than by 20 in right so
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basically they want us to write and
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solve an absolute value equation to
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determine the least and greatest
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possible sizes for the head of this
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racket now here's the thing we you need
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to know what the middle size is right
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the middle size or the mid size is most
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Rockets how what's the surface or the
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area is 100 in square right they can
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either be 20 more or 20 less does that
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makes sense okay they give be either 20
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less or 20 more now common sense to
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say the average brackets between 80 and
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120 in right that's what come say
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because you either less than 20 less
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than 100 by 20 in squ or more than 100
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by 20 in square does that make sense now
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have you change this into the absolute
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value equation this is what we are
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learning how to do here okay so now the
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X here stands for the least or greatest
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size we don't know what it is we don't
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know so this is why we call it x c is
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called a central value right this is how
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you build an act value equation and R is
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the range the range is this we are
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between what 20 Ines less than that 20
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in square less than it or 20 20 in squ
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more than it right so how do you build
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this we don't know X this is what we
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looking for but do we know the central
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value yes what is the central value here
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100 what's the
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range 20 because can either be 20 less
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or 20 more right so now in this case we
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can just build our equation and this is
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how you build it it's X minus 100
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absolute value equal 20 does that make
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sense
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sh are you
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following you following me yes yes all
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right so now how do I solve this
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remember what you told me earlier when I
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have this how do I solve this
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now how would I solve this sh what how
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would I solve this you plug the 20 x no
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we don't plug the 20 into the X
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we have we have to split this just like
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you split it the first time remember
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what you told me earlier if I have
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absolute value of a
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x equals to 8 then X to be what a or X
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could be what 8 right you know what you
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told me so we going to use the same
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principle here if I have value of x - =
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20 I can split it into two right it's
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either going to be x - 100 = what 20 or
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x - 100
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=
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20 okay and then we solve from X from
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here any questions on that everybody get
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it any questions okay and now we can
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solve it so basically the range so a
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tennis bracket a standard could be
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between what 80 Square in and 120 squ in
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does that make
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sense you shake it no you didn't get it
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I don't any of that huh I didn't get any
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of that which one all it all right so
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did you understand what I'm asking
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here no yes or no yeah all right so
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basically you have a tennis racket right
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have you played tennis
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before have you seen a racket before
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yeah all right so the tennis racket
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looks like this and this is called the
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head of the racket right this is the
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head of the rocket so now the problem
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tells us that this head has about 100
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square in of the surface this is this is
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the surface of it is 100 sare in y right
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this is the surface now they're also
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telling us this is PL - 20 in squ which
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means it can be either 100 + 20 in
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square or 100 - 20 in squ does that make
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sense now they want us to write and
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solve an absolute value equation to
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determine the least and greatest
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possible sizes for the head of this
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racket all right so now here's what we
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know we know that the standard racket is
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how how how many inches Square about
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100 are we good now they're telling us
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this is the middle size right it can be
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either 20 in square longer or 20 in
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square less than that does that make
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sense right so if you were to really use
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just common sense to solve this this
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least size will be what 80 in squ
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because this is 100 - 20 and the
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greatest size will be 100 + 20 which is
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120 in s are we good on that now the
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question is that's not what they want us
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to do they want us to transform this
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into an absolute value problem okay and
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now the absolute value this is the
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standard formula for the absolute value
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so it's always when it comes to this
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type of problem
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x - c = to R now what is all this C is
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called the central value your central
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value which is this here all right Matt
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are you seeing this now the range is how
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far can you get away from the central
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value in this case it's either 20 in
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more or 20 in less right and then this
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is telling us if you get away from the
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central value 20 in here or 20 in down
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what would be the value of your bracket
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and this is what we don't know X and
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that's what we looking for and does that
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make more sense okay so now we started
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by saying we know x - 100 = 20 now we
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have to solve this problem we could have
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solved it we could have solved it
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mentally or visually by using what we
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have here but they don't want us to do
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that they want us to actually solve the
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problem algebraically right so now
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here's the thing sh told us earlier that
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if I have absolute value of x = to a it
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means that X could be a or X could be -
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A that's just a standard also if you did
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Val in one or right now how do we use
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this to solve this problem we going to
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use the same method right so now we're
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going to say that either x - 100 is
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equal to 20 or x - 100 is = -20 and now
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to solve each equation separately does
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that make sense so now we going to add
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100
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here and we get x = 120 or we going to
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add 100
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here and then x = 80 so now our
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conclusion
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is the least and greatest possible sizes
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for the head of this bracket is between
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80 in squar and 120 in
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squ does that make sense now are we good
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all right so now now we can solve this
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problem here can I eras
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this now M since you asked the question
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how would you how you going to solve
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this problem this is just said x + 12 =
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9 so how would you solve
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this subract before we subtract what do
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we do first how many cases do we have
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what how many cases we have two right x
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+ be equal to
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what or x + 12 be equal to
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what 9 and then we solve
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it you see what I'm saying now so it's
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two cases x + 12 = 9 or9 the reason why
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this is it because we don't know the
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value of x so because we don't know what
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x is we have to have our two options to
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solve for this you getting it all right
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so now you can solve it okay so go ahead
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and then do this for me I'll give you
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another one to work
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on
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uh you guys look half dead today I don't
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know why it's one day it's a long been a
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long day y + 5 = 13 so go ahead and
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solve this for me all
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right I think we're good with