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How To Solve Inequalities & Graph Solutions On The Number Line
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Feb 19, 2025
In this chapter, we'll talk about how to solve inequalities and represent them on the number line. Chapters 00:00 Introduction 00:00 How To Solve one step equalities 01:28 How To graph the solution on a number line 03:24 Multiplication property of Inequalities 06:01 Division property of Inequalities 06:55 Solving Inequalities by Multiplication or division 08:51 Solving multi step inequalities 11:06 Working on a real life problem involving inequalities
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0:00
so we're going to talk about solving
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inequalities all right again this is
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chapter is still we still on the easy
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side of things so solving inequalities
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if you've done it in algebra one even Al
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pre-algebra this should be just like a
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walking apart it's not that hard okay
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walking apart so the first thing I want
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us to discuss is we're going to first
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solve one step inequalities right one
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step means one step you don't have to do
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too much
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okay we're going to learn how to solve
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it and then we also going to learn how
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to graph the solution on a number line
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all right so how do you solve this
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equation here we have y - 6 is less than
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3 if I want to solve it I'm going to
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solve this just like I was solving the
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regular equation right tell me what do I
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do
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here I'm going to add six to the either
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side right so I get Y is less than 9 now
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the difference between this kind of
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inequality equation is y is less than 9
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here so it's not just a number you want
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the number line a lot of things can
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satisfy this inequality right give me an
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example a number that's less than
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n negative
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what2 yes sir um s seven eight right 8.9
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99 a bunch of numbers so to graph the
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solution we have to draw a number line
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right this is our number line line and I
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have to locate nine nine is here so I
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want everything to be less than nine am
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I going to go left or
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right left right but the thing is this
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I'm going left but there's also
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something else I need to make sure I do
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right you guys know about the closed
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Circle and the open circle right because
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nine is not included is he have does he
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have to be a closed or open
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circle open open is they we're
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comfortable with this open Circle right
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so it's going to be from
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9 open circle to Infinity right this is
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called this is on the number line now if
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you want to represent this as a interval
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it's going to be Infinity to 9 we are
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not there yet so I'm just giving it to
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you as a of knowing okay but this is
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what we have negative Infinity to 9 and
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9 is not included so I put the
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parenthesis not a bracket okay so what
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if I change this problem and I had less
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than or equal why what would be
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different in this case
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yes you'll be a close Circle right
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because now 9 is included if I change
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this to less than or equal 9 will be
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included and I'm still going to be going
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to Infinity negatively but nine is
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included and now instead of parenthesis
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what do I
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have I have a bracket this is important
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right A lot of times people get hung up
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on this they forget to put the bracket
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they put parenthesis that's false
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because if you put par is you're saying
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that n is not included so you don't have
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the same set you have to put a bracket
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right so the one thing that you need to
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remember is this closed Circle closed um
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bracket this
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open that implies a pareses that implies
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a Clos bracket all right so now we going
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to learn about the multiplic
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multiplication property of inequalities
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again very simple step now put a
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statement there 3 is less than 5 we
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agree yes or no unless you live in a
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different Universe three is always going
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to be less than five right now I've
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heard of someone that say that he he
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knew someone that conceived of a squared
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Circle A Squared Circle squared
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Circle yeah can you think of a square
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circle happen right so three three is
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less than five do we agree right now
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what if I times this by -2 and -2 right
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and I write this here
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-6 is less
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than - 10 is this true
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why there a negative right so what do I
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need to do in order to make this
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inequality true yes sir with the sign is
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flip it right anytime you multiply by
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negative we have to make sure we flip
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the inequality
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position right anytime you times by a
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negative you have to flip it it has to
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change because in negativ -6 is greater
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than -10 okay that's just a must now 12
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is more than -7 same thing if I times it
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by-1
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right what am I going to do I'm going to
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get -12
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is what less than postive 7 you have to
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flip it okay and the rule applies also
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for
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what why did you
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use I just used them I just chose this I
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just wanted to does it have to be any
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speci number no I I mean I'm just I'm
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just saying I just this is an example
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right yeah what be7 and not
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where what you
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talk seven because it's 7 * 1 I'm
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multiplying is a
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multiplication this
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one 12 -2 is less than 8 yeah that's a
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completely different problem yeah that's
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true oh I'm saying this is a different
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problem all right this is this is
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totally different okay now this is the
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division property of inequalities right
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I have -12 is less than 8 right if I
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divide this
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by1 what do I get what's -12
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or1 yes 12 12 do I leave this like this
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obviously not I have to change it right
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that becomes eight so again with the
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division property of inequalities
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anytime you divide by a negative you
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have to flip the sign that's just a must
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right so this is why I'm introducing
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this here so if you divide by negative
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or you multiply by negative and
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inequalities are involved the the next
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thing to do is to make sure you flip the
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sign I've seen this too many of too
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often people leave signs like this and
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oh I just forgot so remember that this
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is why I'm recalling this rule okay same
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thing here now we're going to solve
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using multiplication or division here
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right I have -4x this is an example
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solving using multiplication or division
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-4x is less than or equal to 24 how do I
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solve this for X um Co what do I do
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here4 less or equal to 24 divide div4
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and then what switch the sign switch the
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sign so we get
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X is greater than what or equal to -6
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right if I put this on a number line if
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-6 is here where do I go go uh
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right or left right right and then what
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else do I
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do close or open close right because
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it's more than or equal okay now I have
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-1
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/x -1 /
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2x is more than5 how do I solve for x
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here
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kay I don't know what you're asking to
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it right
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here yeah but okay don't know that I'm
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solving for x so what do I do I want to
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get X yes sir
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um oh what else can I do instead of
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dividing you multiply by
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what -2 right the reciprocal right NE -2
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and then this cancels out so now I get
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one or X is what less than 10 again do
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not forget to make sure because you're
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dividing by a negative you multiply by
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negative you have to flip the sign and
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as you put this on the number
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line negative positive 10 is here open
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circle we're going here okay this is
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negative Infinity to 10 open circle now
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on the last one the very last one we are
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solving multistep
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inequalities is Mary missing today Mary
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and Stephanie are both gone
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okay and GG when do the leaves change
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oh I don't know fall season right is
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fall I don't know probably at the end of
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the month I don't
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know so now I'm trying to solve for x
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here so what's my first step this is a
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multi step right what do I need to do
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first times by
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six right on both Sid do you like to
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flip the
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sign no right so I got
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-4x is less than or equal to 5x + 58
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what's the next
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step before we divide we have an X here
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so what I do subract subtract the 5x
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right so I get -
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29x is less than or equal to 58 then
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what's the last step
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now you have a -29 / -29 and do not
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forget
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to flip the sign all right you have to
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flip it so what's 29 into 58 two two so
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this going to be -2 number line going
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from
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-2 to positive Infinity all right so
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this is what this section is all about
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so now what I'm going to do is I'm going
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to give you guys some work to do and
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then we're going to call it a day okay
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so let's do that mhm so why when you
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multip them by six why didn't you put
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the sign there
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it's not netive because it's not
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negative it's positive
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6 very negative right only when you
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multiply by negative or divide by
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negative do you still want your video on
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this or something uh Yes actually let's
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do one word problem and then we going to
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after that we we're going to start
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um I got one more problem after that
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I'll give you all some work to do
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so
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Enrique
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pays uh
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Josh to
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advertise
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on his website
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now Josh's
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website
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earns $15 a
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month
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plus
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0.05 every
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time a visitor
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clicks on the ad right the question is
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what is the
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least number of
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clicks per
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month that
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Josh needs
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in
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order to
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earn
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$50
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month all right so how would you
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translate this problem that's an
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inequality you guys I don't know if you
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know a lot about websites cuz we people
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always ask me what I do I say I'm a
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writer I write on my website I have four
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no five websites that I run right one of
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them is for shoes and stuff like that
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and then people say how do you make
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money on the website you make money from
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ad revenues okay people add on your
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website you make money from there this
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is called passive income you write an
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article once and then people are
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clicking reading the article ODS are
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showing if they click on that you get
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paid so some website makes a gang lot of
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money YouTube is sitting in billions I
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mean probably a lot of money right
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because this is how to make money now
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this guy Enrique he pays Josh to
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advertise on his website maybe Enrique
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is selling whatever maybe earrings may
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maybe he's selling fidget Spinners I
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don't know what he's selling but he
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wants to get more what more exposure
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right so he's paying this guy to
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advertise on his website and joshes
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earns about $15 a month plus
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$0.05 every time somebody clicks on the
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app right so a th000 people click on the
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ad he makes about probably not that much
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$50 right so the question is what is the
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least number of clicks per month that
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Josh needs in order to earn
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$50 so how would you set that up Jack
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let's find an inequality in order to
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earn the least is the key word here
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right least least implies
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what what I say least minimum minimum so
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it's something or more right so how much
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money does you earn a
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month 15 15 we start with that right $15
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plus
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what 0 0.05 since we don't know how much
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uh how many people are clicking we're
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going to put X right and how much are we
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trying to earn at least what 50 so at
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least how am I going to put write that
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uh greater than or equal greater than or
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equal to 50 50 and now we need to solve
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for x right so we need to know how many
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clicks does he need to earn that 50 so
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we have to subract the 15
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first so we get 0.05 X is more than or
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equal to 35 and then we divide this by
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05
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05 so do that in your calculator tell me
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what you
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got
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yeah how much so and that crazy so he
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needs 700 people to click on this in
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order to make at least 50
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bucks guess what if you do that what
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happens if you yourself click on those
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ads this is called this is Shady stuff
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so Google going to find out and you
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going to get banned so people have done
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that people tell other people click on
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those ads they know cuz they have ways
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to figure it out right so this has to be
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people that you don't tell to do that
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it's just people that are coming you
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have to have genuine visitors but the
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fee that he charges is just way too
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little he need to charge a little more
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CU 5 Cent is nothing right cuz you you
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have to get 700 clicks to only make $50
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mean for 500 you need at least 7,000
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clicks right and for for 5,000 you need
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70,000 clicks on that good luck that
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ain't happen so you need to charge more
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money okay all right so that concludes
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that lecture so to speak but now we're
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going to work on some individual
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problems that
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