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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.

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  • leaf green style avatar for user s1371008
    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.
    (33 votes)
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    • starky sapling style avatar for user Angelo Ayala
      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
      (4 votes)
  • marcimus purple style avatar for user Evan Evan
    I worry about Diana's health. 5 biscuits per frisbee is way too much! She's probably eating over a hundred biscuits every time they go to the park.
    (3 votes)
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  • leafers ultimate style avatar for user BrownEMAN15
    so would i write "at most" as less than, greater than, less than or equal to or greater than and equal to?
    (1 vote)
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  • piceratops ultimate style avatar for user Denis Orlov
    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)
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    • starky sapling style avatar for user Sanya Singh
      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!
      (4 votes)
  • blobby green style avatar for user Aisha
    I'm trying my best to understand but the graph is making me confused, is there another that it can be explained.
    (1 vote)
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  • aqualine ultimate style avatar for user Yolie Reyes
    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?
    (1 vote)
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  • starky sapling style avatar for user Amalzakelsherif
    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?
    (1 vote)
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  • blobby green style avatar for user Arbaaz Ibrahim
    I don't understand what the last question in this example was trying to illustrate?
    (0 votes)
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  • mr pants teal style avatar for user Ira B.
    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)
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    • mr pink green style avatar for user David Severin
      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.
      (0 votes)
  • aqualine ultimate style avatar for user Nitkarsh Chourasia
    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)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (2 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.