Luis receives a gift card worth $25 to an online retailer that
sells digital music and games.
Each song costs $1, and each game costs $2.
He wants to buy at least 15 items with his card.
Set up a system of inequalities that represents this scenario
and identify the range of possible purchases
using a graph.
So let's define some variables here.
So he's going to buy music and games.
So let's let m equal the number of songs he buys, and let g
be equal to the number of games.
Actually, just so it's more intuitive
let's let s be the number of songs.
And let's let g be equal to the number of games that he buys.
So they give us some information here.
They say that he wants to buy at least 15 items with this card.
So that means that the total number
of items, the number of songs plus the number of games,
have to be at least 15.
So this sentence right here tells us
that s plus g-- that's the total number
of items-- it has to be at least 15.
So it has to be greater than or equal to 15.
It can be 15 or larger, it could be
16, 17 18, so on and so forth.
It also tells us that he has a gift card worth $25,
and each song costs $1 and each game costs $2.
So he can't spend any more than $25.
So how much is he going to spend as a function
of the songs and the games?
So each song costs $1.
So he's going to spend the number of dollars
times the number of songs.
So if he buys 10 songs is going to spend $10.
Plus, for each game, he is going to spend
$2, plus 2 times the number of games.
So if he buys five songs and five games,
the five songs are going to cost $5, 5 times 1, plus 5 times 2
for the games, plus 10.
So it's going to be $15.
So this is how much he spends on songs and games.
I could have put a 1 out here for dollars,
but this is how much he spent-- or 1 for it's $1 per song.
But this is how much he spends in dollars on songs and games.
And he only has a gift card worth $25.
The total amount he spends has to be less than
or equal to $25.
He can spend exactly $25 if he has to.
He just can't spend $25.01.
So we have a system of inequalities here,
and now we can essentially graph them.
And so to graph them we just have
to pick one something to go on the vertical axis
and the horizontal axis.
And actually I'm only going to draw the first quadrant here
since he's only going to buy a positive number of songs
and games.
So let's let the songs on the vertical axis,
and let's let the games on the horizontal axis.
And then what I like to do is, I like to express it
in terms of-- or if I'm putting s as the vertical axis,
I like to solve for s, so it kind of
goes in the traditional slope intercept form.
So this first thing up here, if we subtract g
from both sides of this, we get that the songs
have to be greater than or equal to negative g plus 15.
And then this one over here, if we subtract 2g
from both sides-- I'm going to do it in that same magenta
color-- if we subtract 2g from both sides,
so that we isolate the s on the left,
we get s is less than or equal to negative 2g plus 25.
And now we can try to graph these,
and we could just look at points.
Or we could look at the slope intercept of the boundary line.
So over here, if he buys no games,
then s is going to be-- well just think about the boundary
line here.
So the boundary line for this constraint over here
is going to be s is equal to negative g plus 15.
So if he buys no games he will buy 15 songs.
So let's make this 5, 10, 15 songs.
So the s intercept, we could say, is 15.
If he buys no games he'll buy 15 songs,
based on that constraint.
Now what happens if he buys no songs.
If he buys no songs, then you have
0 is equal to negative g plus 15,
or you could say that g is equal to 15.
So if he buys no songs then he's going to have to buy 15 games.
So it's 5, 10, and 15.
So that would be a point on the boundary line.
And we're actually going to include the boundary line,
because it's greater than or equal to.
So I'll draw a solid line here.
So this is our boundary line, but it's not just
the boundary line.
It's not s plus g is equal to 15.
It's s plus g has to be greater than or equal to 15.
Or s is greater than or equal to negative g plus 15.
So when you look at it this way, you see s for any g,
negative g plus 15, is going to hit the line,
or it's going to hit right at the boundary line for the s.
But it's that s and any of the s's
that are greater, so we're going to be above the line.
And if you just want to make sure
you can verify that we're above the line.
You could try something that's on the other side of the line,
maybe 0, 0.
0, 0 definitely doesn't work.
0 plus 0 is not greater than or equal to 15.
And if you try something out here like 15 comma
15-- 15 plus 15 is 30, which is greater than or equal to 15.
So we feel pretty good that it's this whole area above the line.
And now let's look at this second constraint.
The boundary line is s is equal to negative 2g plus 25.
So once again, the s intercept, or if g is equal to 0,
s is equal to 25.
So we go to 20, and then 25.
If g is 0, s is 25.
And then if s is 0, what is g?
If s is 0, we get 0 is equal to negative 2g plus 25.
We can add 2g to both sides.
You get 2g is equal to 25.
You can divide both sides by 2.
You get g is equal to 25/2, which
is the same thing as 12 and 1/2.
So if s is 0, g is 12 and 1/2.
Something like this-- 12 and 1/2 would put us-- well
it's actually going to be-- that's 5,
so 12 and 1/2 is going to be right in between these two
guys.
It's going to be right over there.
And once again we're including the boundary line
because it's less than or equal to,
so we'll include the boundary line.
So I'll draw a solid straight line, just like that.
That's supposed to intersect right over there.
And then you can think about it two ways.
You could just say, well, for any g, negative 2g plus 25
will put us on the boundary line,
and we want all the s values that are less than that.
We are going to be below this line.
And you can verify it.
Take some points that's on either side of this magenta
line, 0, 0 should work.
If you have 0 plus 0, that is less than or equal to 25,
so that works.
15 comma 15 shouldn't work.
15 plus 2 times 15 is 3 times 15, so that's 45,
and that's not less than or equal 25.
So stuff out here above the magenta line won't work.
Stuff below the magenta line will work.
Now the range of combinations of songs and games
that will actually work are the overlap
of these two inequalities.
The way we've visually drawn it right here,
they only overlap right over here, this region
that I'm shading in here in blue.
But it will include the boundaries
because both of those constraints
were-- included the equal.
This is greater than or equal to,
this is less than or equal to.
So it will include the boundaries.
So any combination of songs and games
that fit in this boundary right here
will satisfy his constraints.