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## Modeling with linear inequalities

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# Graphs of two-variable inequalities word problem

CCSS Math: HSA.CED.A.3

## Video transcript

- [Voiceover] Diana the dog receives five dog biscuits for fetching each frisbee, and three dog biscuits
for fetching each ball. Sounds like a pretty good deal. She plans to receive at most D dog biscuits before chasing her tail. Well, that's sounds reasonable. The inequality graph
below represents the set of all combinations where
Diana fetches F frisbees and fetches B balls in order to receive at most, at most, D dog biscuits 'cause at that point she
reasonably starts chasing her tail. According to the graph, we're gonna take a look at the graph in a second, according to the graph, what
is the most number of dog biscuits Diana wants to receive
before chasing her tail? In other words, what is D? So let's look at, let's
interpret this graph properly. So if we look at the horizontal
axis right over here, that's F, that's the number of frisbees, the number of frisbees
she catches, frisbees and this vertical axis,
this is the number of balls, number of, number of balls that she gets and we know what the total number of biscuits are going to be. The total from catching frisbees,
if she catches F frisbees, she gets five biscuits per frisbee so the total from catching frisbees is 5F and if she catches, if she catches B balls or retrieves B balls so
if she gets those B balls, if she gets, what was it? Three? Three biscuits per ball? Yep! Three dog biscuits for fetching each ball, the total number of dog biscuits she gets for catching B balls is
3B and so the total number of biscuits she fetches is 5F plus 3B. This is the number of
biscuits from frisbees. This is the number of biscuits from balls. Now we can see all of the
allowable combinations of number of frisbees
and number of balls here. And so for example, if she catches that point right over there, eight or retrieves eight balls and catches, well, that would be half of a frisbee, so that doesn't, that
doesn't seem to make sense but if she retrieves eight
balls and catches one frisbee well, then, that's still
ok, she still hasn't met her maximum number of biscuits yet. So how do we think about the
maximum number of biscuits? Well, the maximum number of biscuits are any of these points when she, that are sitting on this line
and notice, the solutions that where all the points
it satisfies this inequality are all below this line
so she's hitting a maximum when she's on the line and an easy one might be this point right
over here where we see that, where we see that frisbees,
zero frisbees and 10 and 10 balls pretty much
maximizes her number of biscuits. So if she catches 10 balls,
so let me write this down, so if B is equal to 10, B is equal to 10, F is zero, if F is equal to zero and B is equal to 10, well,
how many is she going to catch? Or how many biscuits is she going to get? Well, she's going to get,
this is gonna be zero and then three times 10 is 30, so that's gonna be 30
biscuits, 30 biscuits. So this point right over
here, this corresponds with 30 biscuits, 30 biscuits and you can see that any of these points along this blue line, actually
correspond to 30 biscuits. If you go over here, where F is six, let me write it here, F is equal to six and B is equal to zero so
spends all of her time, she's earns her biscuits purely through frisbee catching,
so this is a situation where F is equal to six
and B is equal to zero, you still have the same scenario, you still have, if F is
six, five times six is 30 plus three times zero,
we'll that's just gonna be once again, 30 biscuits, 30 biscuits. So her maximum, or the number of biscuits she needs before starts
chasing her tail is 30. So D is going to be 30 and
in fact, we can express this inequality as a 5F plus 3B has to be less than or equal to 30. Alright, then they ask
us another question. Can Diana fulfill her plan by fetching, can Diana fulfill her plan by fetching four frisbees and two balls? So let's see, four frisbees and two balls, this is right over here,
four frisbees and two balls. So it's not, we're not saying that she has to maximize, that she
has to get the 30 biscuits, she just cannot eat any
more than 30 biscuits. So it seems like she can
fulfill her plan, let me see. The inequality graphed below
represents all the combinations where Diana fetches F frisbees and B balls in order to receive at
most D dog biscuits. So let's see, is she fulfilling
her plan by fetching, well, yeah, I would say, yes.