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### Course: Algebra (all content) > Unit 6

Lesson 4: Modeling with linear inequalities- Writing two-variable inequalities word problem
- Interpreting two-variable inequalities word problem
- Solving two-variable inequalities word problem
- Graphs of two-variable inequalities word problem
- Two-variable inequalities word problems
- Modeling with systems of inequalities
- Writing systems of inequalities word problem
- Solving systems of inequalities word problem
- Graphs of systems of inequalities word problem
- Systems of inequalities word problems
- Analyzing structure with linear inequalities: fruits
- Analyzing structure with linear inequalities: balls
- Analyzing structure with linear inequalities

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# Graphs of two-variable inequalities word problem

Given the graph of a two-variable linear inequality that models a context about dog biscuits, Sal finds if the dog can get enough biscuits.

## Want to join the conversation?

- How is she fulfilling her if she wanted to receive the MOST possible dog biscuits? because 4 frisbees and 3balls would mean only 29 biscuits.(34 votes)
- she does not want to receive the most dog biscuits, she wants to receive as most 30 but any below that is also okay. as the point you said is not on the line but below it she would not get the maximum amount as with any other point below the line. again as the equation to this is 5F + 3B ≤ 30. 29 works(11 votes)

- I do not think Diana fulfilled her plan. At (4,2) Diana still had NOT received MOST biscuits, therefore not fulfilling her plan.(0 votes)
- Diana does fulfill her plan because she wants to get AT MOST 30 biscuits, not more than that. She is okay with getting some biscuits less than 30! The point (4,2) is in the shaded area which means she is fulfilling her plan as she has less than 30 biscuits.

5F + 3B ≤ 30 => F= 4 & B= 2 => 5(4) +3(2) ≤ 30 => 20+6 ≤ 30 => 26 ≤ 30!

So, yes she does fulfill her plan!(15 votes)

- so would i write "at most" as less than, greater than, less than or equal to or greater than and equal to?(1 vote)
- Less than and equal to(3 votes)

- Bruh. Where did the 30 come from?(1 vote)
- She gets 3 biscuits for every ball, so after fetching 10 balls and no frisbees, she'd have 30 biscuits(4 votes)

- ERROR in my question of 5 minutes.

I made an elementary mistake as the equation is NOT :

B= -5/3F +D but

B= -5/3F +D/3

Correction between angle brackets (<>) has been inserted below in my original question :

"From the problem "wording" I can derive the Diophantine equation D=5F+3B or B=-5/3 F + D</3> which corresponds to a bunch of parrallel lines with negative slope -5/3 and y-intercept= D</3> and D still unknown. So how can I from these incomplete data derive the graph? From which data did Sal derive that the y-intercept equals 10 <so D=30> rather than any other value?"(1 vote)- The graph was given as part of the problem. So, it did not need to be derived. Without the graph, there isn't sufficient info to determine the graph for the line.(1 vote)

- I'm trying my best to understand but the graph is making me confused, is there another that it can be explained.(0 votes)
- Is the reason we don't say the maximum number of biscuits the dog can get is 5(6)+3(10)=60 is because the graph connects between the points (0,10) and (6,0)? if it included the point (6,10) we could've said the dog can get the max no. of biscuits for both the balls and Frisbee and that would be 60, right?(0 votes)
- Even though this seems like a real life word problem, it really isn't because dogs can't do the math. What actual real life word problems would work for inequalities?(0 votes)
- The ones about going to fair and you have 20 dollars to spend, it costs 5 dollars to get in, some rides are 25 cents and some are 50 cents. If you want to ride at least 10 rides, what are some possible combinations of 25 and 50 cent rides?

Anytime you have a limited amount of money to spend, you can figure out what you might want to buy.

This can get bigger scale if you are talking about a company and the amount of inventory you might want keep on hand to maximize profits.

With launching rockets, you have to balance how much fuel you will need with the weight to launch the rocket.(2 votes)

- How we should know that x-axis is the number of frisbees and not the number of balls or vice- versa how to know that. How he assumed that. *Can please anyone help?*(0 votes)
- First - the graph was given along with the text description of the problem. You need to read word problems very carefully. It pays to read them more than once to make sure you understand what info you are being given.

In the 2nd paragraph, it tells you the F=# of frisbees and B=# of balls. So, then look at the graph. The x-axis is labeled with "F". So, the graph is telling you that it is number of frisbees if you know that F=# of frisbees. Similarly, the y-axis is labeled "B", so it tells you that is it number of balls.

Hope this helps.(4 votes)

- I wondered how she would fulfill her plan of reaching D, when we found out that D is the maximum number of dog biscuits equal to 30. Original equation: 5F + 3B = D,

F=4 and B=2.

5(4) + 3(2) = D

20 + 6= D

26=D

However we already know that D is actually equal to 30 using the graph so

26=30 is not correct, and so does not satisfy the inequality. And so the answer would be no because her plan of getting D biscuits was not achieved.

Or am I not understanding this?(0 votes)- It only means that anything
`<=30`

is OK. Even 3 biscuits are OK.(0 votes)

## 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.