today we talk about sex and the
intersection one point six and one point seven we talk about
three different things first we're gonna talk about my god
era equations that are quadratic inform second we're gonna talk about
Lennar Lanier equality any qualities and then finally absolute values we're not
talking about absolute value and equality we're talking about
absolute value equations go cry about that alright so first thing we can talk about
our equations that we can describe as being quadratic
inform so tell me what what something
be if it was quadratic inform what would it look like but for things four times right so well that first term look
like if it was quadratic inform something with a square guy in
may have a number in front but it's got to have something that
squared right what's that second term
gonna look like something like with an axe or
just the first power and mayor may not have a number in front that their number mayor may not
be there right constant and then equal zero they ask and
I have four pieces right quadratic inform we'll be in general something
squared but that same something to the
first power plus a number plus minus equals
zero quadratic and born so what we're going to talk
about today is equations that aren't exactly
quadratics but they are quadratic in form so keeping that in mind this is all they say not a
quadratic because it's fourth degree right to the fourth but can't we say that it's
quadratic and form yes how would we do that you
really need to put that way it's so irritating to me when people
are texting and I'm teaching kind of a slam to me alright so how could that be
like a quadratic inform what could it sets it teaching
could I make XQuery can I use some number our
value like you is ex squared right and then what would you do
squared equal X. to the poor right so if I let Houston her ex
squared menu squared is next to the four and cannot rewrite this whole
thing what what ari place next to the
fourth with you square right and then minus four times you just you right plus three equals zero now is this a quadratic yeah and that's easy quadratic
that we can solve because y'all are all quadratic solving
experts so how does that quadratic
factor you can you know the way we can
do that right how the only way we can get
three three one Gaby three one if that's positive in the middle
of Times negative I get a bit both be negative
right negative three negative one so we know that's how that
factors so if I was going to say solve
the I would say you think well that's very
effeminate what is the rope right said to me one of their
assault I get you is equal to three and Yale is equal to one so that's the solution of this
quadratic but I would make but what are they trying to
solve to begin with this fourth degree yeah Joey decides it back and exactly so we said that you was that squared right so we just go okay then EC
squared equals three and that's easy so what that plus or minus right plus or minus the square to three and if you as a whole the one but what is you stand for elsewhere it again so if X. squared equals one man
acts is plus or minus one so for solutions plus or minus us where three
plus or minus one so we would write that negative three native run
positive one because the three that makes sense so ideas quadratic informed
something that's not exactly a quadratic but can I make a
substitution said riding as a quadratic saw that quadratic and
then sets you back makes a lot easier yes %HESITATION Yreka call I had it
right there thank you square root if three square root
yes so I have their transferred over do what use that situation is that what
your teacher called yeah that works yeah alright so this time the radical we talked
about radicals last night in making get a little intense on Friday was that last week
Wednesday so we could solve this on that
way if we wanted to write could we also consider met this kid be quadratic inform there was gonna be quadratic and
forum what would you stand for if we let you the the square root of bats then what's you squared just that now you may not have and
%HESITATION identified it easily but it
works so if I make that substitution then I rewrite this dead you
squared minus sixteen you minus five hundred twelve spirit of five twelve I lost it there it but not a script so what if factors of five
hundred twelve anybody know something less than twenty two chief if you think that we want
to get sixteen in the middle that's divided by nine nope but about eight that's not our bye bye for yeah but that will give a
sixteen divided by seven no anybody think he said there is a thirty two does that work okay I just suddenly woke up and
help alright so you're saying you and
you and if I divided by sixteen the
answer is thirty two that work one's gotta be positive ones got
to be a negative so which runs the negative but thirty to the negative miss
on the positive brought it so I found the link here so I get that you is equal to
thirty two and you is equal to negative
sixteen motor easy now what square square root had we know
what you stand for where'd abaixo I set you back
said the squared of axis thirty two so actually do square them
right so square both sides so X. is equal to nine sixty four that's right and then at the square root of
you is equal to negative sixteen van credit at that that's pretty you then access equal to fifty six yeah okay solutions your question I got that wrong NET squared is
not nine six or it's a thousand for okay your question why do we
just fine except do we find what no we just signed acts we're always the thought and
practice if you want to know like squared then you can but we found that the square
root of axis thirty two but we really wanna Gary answer just in
terms of it the question alright emit else event okay you're saying this number five a
lot of fun so access to the faint to
parents plus three X. so the one third manifests border so if you
wanted to write that as something quadratic emporium
which should be the easiest what do you know what you stand
for acts if sex exit the what one third right can't really find a partner you
with your body okay so did we start here and
that we recognize that if you take this term and
square it you get this term okay that's in general how you
gonna identify something it's being quadratic informed if your equation is of that form then you take this term and you
let that stand for you not the three because the you back to
the one part sobering out you is evil the axis of the one third which means if we square that
what happens on you square something to the one third power it's to the two thirds power but you remember that thanks to the one very square you multiply their clients and
so use square it's represented by acts to the
two thirds power not so now we make that
substitution we rewrite this adds use where was three you minus four equals zero we factor it as a factor plus fire you minus one it felt a linear equations so you wake was negative for her you think on the positive one
then what that's the tape back say what do
you see for U. standard for acts to the one third so now we say
okay then next to that one Perry equals negative for an axe to the one thing very
equals one and last time we talked about
what we do if it has a fraction for an exponent you take it to what power the reciprocal right so we're gonna take both sides
here we're going to Cuba because if
if that one very power and we keep it including the negative so it's a
negative for here too so what's next gonna beat negative sixty four right
because negative times a negative times a negative is
still a negative and if we keep you one we still get one okay as one cube is mine let us have fun you know alright so that's how you do
ones that are quadratic inform okay alright the second had the thing
we talked about today absolute value equations we won't cover
absolute value inequalities so remember that you know that if we're talking about
absolute value it's defined as the distance
from acts this is a bad from vero on a number line I see
it definition what's in absolute value of negative type thing nineteen so if I ask you to solve that's a
value of X. is equal to eight I'm thinking out some members
who that's a value is eight could be a what I was going to
be evade camping brawl with ray yeah so we say that the solution
here is acts as a positive thing or axes and negativity both of those are still she alright so in general if for solving
some absolute value of you is eager to see the answers are so
presidency and negative right athlete I and do you know
that the absolute value graph okay this I think about it all the
positive numbers Nestle valuable positive one is one that's
invaluable positive to you is to write but what about the
negative numbers what saps the value of negative one one without survey of native to
the two together that's why the graph looks like
a B. that's why if I'm asking you what it where is the
absolute value whatever that number is always happen that you places right the positive and the
negative how it's okay so absolute value are
interesting so if I ask you to solve where
is the absolute value of two X. minus one is equal to five what do you tell me if they can happen at two
different places isn't going to happen where to
act minus one is a positive five and two X. minus one is a negative
factor two places that happens trip so if I thought of those add one to both sides I find
that you act as if with a six Alexis equal to three and one of us right here to access the negative forest so
access is a negative to so my solutions are native to
and positive three alright your turn are they going to the glory
because you're looking bored Hey actually only rose to and for
everybody else is gonna work recruits because only two but before you go think about this so the absolute value is equal
to another absolute value yeah could you just say this
expression is equal to the positive of that and the negative of that network yes okay alright everybody working
groups either on the border with your
buddies so let's see did we solve this
one by first setting one half tags plus two equals two three fourths acts
minus two and then that we all felt that
one half X. plus the two equal to negative three four acts mine is to every start there JC really grab it from my I and then some of you love
fractions that for those of you who don't I would highly
recommend getting rid of them by multiplying everything by four
if you didn't do that on it until the fractions how good
time that but you don't have to I think we did we we get you
acts was eight is it well the three axis minus
eight it probably looks a lot easier
than some of your stand I'm so glad subtract two X. from
both sides subject three acts for both
sides I'm sorry the facts you actress and at eight so X. is equal to sixteen one
solution and then the other one over here we have one half axis plus the
two and we want to distribute that negative to both terms sound negative three for X. plus two and again I would highly
recommend multiplying everything by four
because then we don't have fractions so we get you access plus eight equal negative three acts plus eight so if we subtract X. from both
sides are eight from both sides and add three acts we end up with five X. is equal
to zero which means what access to be zero so our solutions where the raw
and sixteen I cannot check that answer do you guys know how to use an
absolute value function on your calculator let me show you and Sir you go to mass and over to number and then
function one for absolute value okay that's the value of it one
happened point five so point five of sixteen was to on point five two nine sixteen plus two several FIDS time and then the right side if I go
to Matt absolute value again so set point seven five for
three four hours I am sixteen mine is to both sides are gonna ten so
Reagan number zero can do the same
thing if you want to see what that look like a crazy
graphically that kind of interesting can do the same
thing welcome mat number absolute value but grab a point
five X. plus two and then the graph %HESITATION point seven five X. minus you it's kind of fun just look at
the graph so there is the first time where is the second one still you can see that they neat
way out here at sixteen if you made the window wider you'd be
able to see that if you go to affect an end table
you looking for where the equal they're equal to each other one
axis zero and then if I went down to
sixteen very close to each other again so lots of different ways that
you can check your answer many questions about that
absolute value alright how many of you remember
that from how rich you are maybe even as everyone now you should have done it but
maybe I got somewhere hi so the last thing we talk about
today are solving any quality which isn't a big deal you've
done it before and the thing that we're gonna talk about that
may be a little different is how to describe this solutions and
turns as %HESITATION interval notation just print
maybe anything for you so we know that we described the
solutions to in any quality in terms of a set of numbers right
there's not just one number that makes this trip probably infinitely mean many
right so we describe them in terms of a set so when we solve
an inequality it works pretty much like an equation the only thing that's different my more working less and any
quality versus equation is if you ever had to divide or
not the pie by negative what happens if you change the direction of
the inequality that's the only thing that's
different at one whatever we work with an
equation bridges and equality as far as procedurally alright so we would get four
axes left and twenty four divide by four sell ads is less than
six but how do we describe that
solution okay where he talks but it's a
lot of numbers I was gonna make a number line
in here's the number six then where are all of the numbers that are the
solution right on that site but is it six part of that solution yeah so it's everything less
than six so some people learn to put it
open circle there how many do you learn to pet a
parenthesis some of the okay where guarantees a round print
is not a square one no the bracket apprentices and that includes that means an
open interval mean that does not include that number okay so from six and we were all
the numbers this right Lotto numbers in that group
right how many and finitely many okay it was established to describe
that in terms that set notation I would describe this same group
of numbers something like this little squiggly bracket and I
say antics and then a line that line means sites that that's what it means axe sex that ads if less than six squiggly brackets and that would but what is
called set notation and you've done that before
probably on a paper but when you have to type this
in a computer it gets a little irritating okay so I don't know if it's because of
computers are right that we have something called interval
notation in an interval notation we can
describe the same set of numbers the number line set notation we
can describe it really precisely as just being everything from negative infinity is six
with around wrapped up print this that means the same thing okay and were inside about that
more but this is and travel not Asian and I'll give you the
details in a minute kind of messy now given that any of the six alright so let's do another one so if I'm solving this one what
I do first that chat theory seven negative five acts is
greater than or equal to four so what do I do divide by negative Phi which
means big Big Bang DR fits in any quality at divided by
negative or if I multiplied by negative I changed the direction that was greater than not equal
now it's less than or equal to negative four fifths okay Steph I'm describing all of
these numbers my number line and I say here is negative poor
fits do I include that number yes how do I know that because it's also equal to its
participation and if I'm looking for Allah and
members less than maybe four fifths which way do I go to the left so I'm gonna say my
solution and because I want to include that number at the
square bracket some of you you stupid like
color color then circle right square bracket same as the code
and circle and I'm talking about everything that way obviously we don't want a job
number line when repent our answers and mymathlab because
that would be irritating so we describe it in terms that
set notation again addicts such that the axe less than or equal to negative
four fifths what the characters to type so much candidates way of doing that if I describe it as all the
members between negative infinity and negative four
fifths and I put a square bracket which means include that
boundary alright so heres the %HESITATION specifics about interval notation how many of you have used
interval notation most of the it's good it's good and that's not a big deal it's
just an easier way kinda like shorthand for describing as a
set of numbers so the open interval means we
use parenthesis that's around it okay that means we don't include
the boundaries so like if I was talking about
all the numbers and the tween five and eight our pet like me this way I
include all those numbers interval notation means just
five com eight set notation and I have to write
accepts that I was less than taxes less than
eight if you're typing in matlab which
one's easier interval notation okay if it's a closed interval meaning I want to include the
boundary you put the square brackets okay so the square bracket it's
what you put to include it cell in this case I would put it
square bracket one comment three with a square bracket because
that one included both found grace can they be mixed up in one
boundary would be included in the other one wouldn't yes so here I just put acts thanks that one is less than or
equal to axes left manner equal to three where you talk about the
employment and they're all meaning everything from the same
quake to infinity are too negative infinity said like this one and actually
this is a square bracket here goes to infinity stuff I was
describing that enable but square bracket starting at negative one going
on out to infinity can you ever get to infinity that's why we always pay around okay an open interval with
infinity or negative and you'll never get there so we can include it so then set notation and that
they be accepts that act is greater than or equal to
negative one and then finally employment and
I will look to the negative %HESITATION direction this set of numbers I
would describe as negative infinity to negative one this is
round okay I'm not including that
boundary though I would make this rounded and that's set notation for that
would just be acts such that acts is less than native one okay so I paid close attention if you're including the boundary what kind of symbol be put by a bracket that square one right
if you're not including that boundary you put the rounds apprentices pay and we always
put the princess's the rounded one by either positive or
negative so that's kind of it we always
talk about an answerable given the lower boundary and the upper okay all rights lower upper on our under number nine was the
left side was the right side so we would never describe this
said the interval between three and one right that that would not make
any sense at all the smaller number has to be
first the lower boundaries first then the upper boundary okay so if I ask you to solve this
inequality and then I ask you to describe
the solution and instrumentation and set notation Hey everybody that wasn't the
Boardman against gets go go to the board now and then everybody
at the desk right together okay so what do you guys see
first and five to both sides we get off at the track or at at
the same time if we wanted to and then you just get acts is
greater than or equal to negative six so on a number line if this is
negative six start here to the right greater than or equal to sell
and answer limitation square bracket everything from negative sixteen
and vanity and round or square goes here round right because we can ever
get to infinity insight meditation and we say acts such
that X. is greater than or equal to negative six I so what about this man what if
there is an inequality on both sites can we just divide by two on
every term yes we can't Sarah gets here is less bad acts
is less than or equal to five so let's shall not relevant to
describe the set of numbers so I'm looking for everything
between two and five do include the two now so I have had their rounded apprentices do I include the five yet so Brackett and we're talking about
everything in between everything in the middle right so in answer will not Haitian
weakest head everything from two to five you know anything is we put
square right here that means include death five
but not that you now the numbers in between and ensure a sanitation or just
be acts such that that that would be the set notation still pretty easy I'd say let's talk about
calculating very they cracks me up yeah so many times in math
class I get students to lateness women can you tell me what my
grade is and I just want to go now you
get to fail if you can calculate you crazy but I won't be rude but that's how I
feel sometimes so let's practice so let's say Claire has received
scores of eighty five eighty eight eighty seven and eighty on
her house protest what school are my she receive
on the fence test so that's the big question well it's going to have to make
if you want your overall test average to be at least eighty
two what's the lowest number at it cracks me up why would you shoot for the
lowest I don't get it do you study rise to make it that's
that's seventy races that you were making eighty two what
what's that magic number I don't it's funny alright so let's think about
this were averaging averaging your scores so what's the first thing we
should do I don't know what this star Adam
up but we only have four there's a fifth one hi what yeah what if we let tags beer fifth test score scorer on this test that's a really big
deal otherwise hate on average them
you don't have that one yet but now that you have about one can I describe my average yeah that's not hard right well now we are not right eighty
five plus eighty eight plus eighty seven plus eighty
plus what addicts what I need to do if I were to
describe the average then I want to divide them by
however many there are right average right so all this score
is divided by five if I wanted to be at least
eighty two at least eighty two what
inequality signed up for it greater than or equal to Brad Pitt at least Sarah greater
than or equal to eighty two why did I put it greater than or
equal happy Madison had greater than well I mean I made more than
eighty two right eighty Jews okay obviously so that's why we say at least
means greater than or equal you can just multiply by five
and be a great thing and Adam all up right so where the scores equal
to so eighty five plus eighty eight
plus eighty seven plus eighty okay three hundred forty plus access over five greater than or equal
to eighty two now multiply both sides by five what's eighty two times but so I know that three hundred
forty plus access has to be greater than or equal
to four hundred and ten now at track three forty so what you have to make at least seventy supervision is a study
like three point two hours right if you're making a deal on a
steady like for how do you know I don't get it I'm just teasing so what if there was something a
little more complicated like ours that have a weighted scale it's not just an average money you ask me or how I Peter
out my grade well look on your course outline and it says that the quiz it's worth ten percent the homework is worth ten
percent the what and let's just say for grants
let's say that exam are worth the remainder because we got some more stuff
but we haven't we don't have any other grades right so far we
don't have any prejudice we don't have anything else so this IBM to give us a way to
estimate okay so let's pretend that you know
that your quiz averages a seventy eight but when you're number average
is a ninety eight and you're worried about mid
terms because you're on a sports team and if you're not passing
you don't get to play what what do you have to make on
this it next exam so you'll be passing just like right now that
doesn't mean the end of semester just right now how right how to
write that out okay that's a thought seventy eight plus ninety eight
plus acts it's greater than or equal to
seventy anyway I think I've your problem it's only a percentage right we
don't add the grades we had a percentage of the greats so in order to make this where seventy eight the quiz it's only
worth ten percent this one is the home are great
it's worth ten percent but this is the big issue that test grade is worth how
much we're gonna say eighty percent their point eight of acts which we don't know over
making heights sex is what we can make amazing we want that to
be greater than or equal to seventy so given that you know your quiz
grade seventy eight your homework grades ninety a this is the equation you that makes sense everybody we don't know but you can do that but it does get more complicated
throughout the semester because we'll have more categories but
right now that's about the only categories we had say later on that we want I thought you knew you had a
question please work here reading this
weekend and I think share anything down
and listen from now and say something something really
really important alright so as you're doing the
review mymathlab tell me what's the purpose of
this review to see if you have ever seen the
written material right so if you own your body work
that's hallmark their review together
you make a hundred but your body does most of it for you what do you think you're gonna
make on the exam that's why anyone not be awesome
right I let me tell you the correct
way to use this review here's what I did I think
basically three and every time a problem you're
going to see okay so if you're working on
this one and you know hide I don't know
how to do this I can't remember it I'm bad at this okay if you did this and the
very first time you looked at it I don't know how to do it I had
to get somebody help me here's what you do you do you
get a piece of paper are no book and you write down this number
one point two point twenty five and you start making a list of all the types of problems
that you need to practice right this is meant to tell you
what you need to practice okay so as you're working you
you make a list these are the things I need to practice and then you practice here's where you practice you go to the study plan and you click view all chapters
otherwise it just show you some previous stuff a well that number the one that
I couldn't do you cell chapter wine section one
and I can either go here or I can just click on Iraq if I knew
what the topic was I could click on it if I just click here all the problems in section one
point one so could I go here and work
again one point whatever it was I was too whatever estimate is
this can I go ahead and work that one and I'm like this one and this
one and this one and this one and this one too because they practice you get
good at it and you'll have the pontiff's that makes sense that's how you use the state
plan this is all the problems in the
whole book or in a study plan everything the ones at the
little video have somebody stupidest guy you do that okay so please use the study
plan well it doesn't affect your grade at all it's just a place
for you to practice okay that's how you use the
review to be about

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