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# Modeling with systems of inequalities

Sal models a real-world context into an algebraic system of linear inequalities and graphs it. Created by Sal Khan.

Video transcript

Luis receives a gift card
worth $25 to an online retailer that sells digital
music and games. Each song costs $0.89 and
each game costs $1.99. He wants to buy at least 15
items with this card. Set up a system of inequalities
that represents this scenario and identify the
range of possible purchases using a graph. And that's why we have some
graph paper over here. So let's define some
variables. Let's let s equal the number
of songs he buys. And then let's let g equal the
number of games that he buys. Now if we look at this
constraint right here, he wants to buy at least 15
items with this card. So the total number of items are
going to be the number of songs plus the number
of games. And that has to be
at least 15. So it has to be greater
than or equal to 15. So that's what that constraint
tells us right there. And then the other constraint is
the gift card is worth $25. So the amount that he spends on
songs plus the amount that he spends on games has to be
less than or equal to 25. So the amount that he spends on
songs are going to be the number of songs he buys times
the cost per song. Times $0.89 times-- so I will
say 0.89-- times s. That's how much he spends on
songs plus the cost per game, which is $1.99 times the
number of games. This is going to be the total
amount that he spends. And that has to be less
than or equal to 25. Now if we want to graph these,
we first have to define the axes, so let me do
that right here. And we only care about the first
quadrant because we only care about positive values for
the number of songs and the number of games. We don't talk about scenarios
where he buys a negative number of songs or games. So just the positive quadrant
right here. Let me draw the axes. So let's make the vertical axis
that I'm drawing right here, let's make that the
vertical axis and let's call that the song axis. So that's the number
of songs he buys. Let me make sure you
can see that. That is the song axis. And then let's make this, this
horizontal, that's going to be the number of games he buys. Let's bold it in. And just to make sure that we
can fit on this page-- because I have a feeling we're going
to get to reasonably large numbers-- let's make each of
these boxes equal to 2. So this would be 4, 8, 12, 16,
20, so on and so forth. And this would be 4-- this
obviously would be 0-- 4, 8, 12, 16, 20, and so on. So let's see if we can graph
these two constraints. Well, this first constraint, s
plus g is going to be greater than or equal to 15. The easiest way to think about
this-- or the easiest way to graph this is to really think
about the intercepts. If g is 0, what is s? Well, s plus 0 has to be greater
than or equal to 15. So if g is 0, s is going to be
greater than or equal to 15. Let me put it this way. So if I'm going to graph
this one right here. If g is 0, s is greater
than or equal to 15. So g is 0, s, 15, let's see,
this is 12, 14, 15 is right over there. And s is going to be all of the
values equivalent to that or greater than for
g equal to 0. If s is equal to 0, g is greater
than or equal to 15. So if s is equal to 0, g is
greater than or equal to 15. So g is greater than
or equal to 15. So the boundary line, s plus g
is equal to 15, we would just have to connect these
two dots. Let me try my best to
connect these dots. So it would look something
like this. This is always the
hardest part. Let me see how well I can
connect these two dots. Nope. Let me see. I should get a line
tool for this. So that's pretty good. So that's the line s plus
g is equal to 15. And we talk about the values
greater than 15, we're going to go above the line. And you saw that when g is equal
to 0, s is greater than or equal to 15. It's all of these
values up here. And when s was 0, g was greater
than or equal to 15. So this constraint right
here is all of this. All of this area
satisfies this. All of this area-- if you pick
any coordinate here, it represents-- and really you
should think about the integer coordinates, because we're not
going to buy parts of games. But if you think about all of
the integer coordinates here, they represent combinations of
s and g, where you're buying at least 15 games. For example here, you're buying
8 games and 16 songs. That's 24. So you're definitely meeting
the first constraint. Now the second constraint. 0.89s plus 1.99g is less
than or equal to 25. This is a starting point. Let's just draw the line 0.89s
plus 1.99 is equal to 25. And then we could think about
what region the less than would represent. Oh, 1.99g. And the easiest way to do this,
once again, we could do slope y-intercept all
that type of thing. But the easiest way is to
just find the s- and the g-intercepts. So if s is equal to 0 then we
have 1.99g is equal to 25 or g is equal to-- let's get a
calculator out for this. So if we take 25 divided
by 1.99, it is 12.56. g is equal to 12.56. So when s is 0, let
me plot this. When s is 0, g is 12.56. This is 12, this is 14. 12.56 is going to be right
there, a little bit more than 12. That's that value there. And then let's do the same
thing if g is 0. So if g is equal to 0, then we
have-- so this term goes away-- we have 0.89s. If we use just the equality
here, the equation-- is equal to 25 or s is equal to-- get
the calculator out again. So if we take 25 divided
by 0.89, we get-- it's equal to 28.08. Just a little over 28. So 28.08. So that is, g is 0, s is 28. So that is 2, 4, 24, 6, 8. A little over 28. So it's right over there. So this line, 0.89s plus 1.99g
is equal to 25 is going to go from this coordinate,
which is 0, 28. So that point right there. All the way down to
the point 12.56,0. So let me see if I
can draw that. It's going to go-- I'll draw
up one more attempt. Maybe if I start from the
bottom it'll be easier. That was a better attempt. Let me bold that in a little
bit, so you can make sure you can see it. So that line represents
this right over here. Now if we're talking about the
less than area, what would that imply? So if we think about it, when
g is equal to 0, 0.89s is less than 25. So when g is equal to 0, if we
really wanted the less than there, we could think
of it this way. It's less than instead of just
doing less than or equal to. So s is less than 28.08. So it'll be the region below. When s is 0, g-- so if we think
s is 0, if we use this original equation, 1.99g will
be less than or equal to. I use this just to plot the
graph, but if we actually care about the actual inequality, we
get 1.99g is less than 25. g would be less than
or equal to 12.56. So when s is equal to 0,
g is less than 12.56. So the area that satisfies
this second constraint is everything below this graph. Now we want the region that
satisfies both constraints. So it's going to be the overlap
of the regions that satisfy one of the two. So the overlap is going to be
this region right here. Below the orange graph and
above the blue graph, including both of them. So if you pick any combination--
so if he buys 4 games and 14 songs,
that would work. Or if he bought 2 games and
16 songs, that would work. So you can kind of
get the idea. Anything in that region-- and
he can only buy integer values-- would satisfy