Graphing Systems of Inequalities Graphing Systems of Inequalities
Graphing Systems of Inequalities
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- 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.
- 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
- 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
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