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Two-step inequality word problem: apples

We'll talk you through this fun and challenging inequality problem. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • female robot grace style avatar for user Ritika
    I still don't get what an inequality is too well. I'm sorry, but can someone explain?
    (28 votes)
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    • old spice man green style avatar for user Skywalker94
      I'm assuming that, since you are currently studying inequalities, that you've probably already done some basic study on the equation, and as you most likely already know, and equation states that one side is equal to another.

      If I have the very simple equation X = 8, it is saying that X is equal with the other side of the equation, which is just 8. That is pretty simple, right? X just has 8 in it to make it equal with the other side. Most people don't really have to think too hard to know that X is just 8 in this case.

      Well, an inequality is just the opposite. It states that the sides ARE NOT equal, and shows how. What if we saw X < 8? This says that whatever is in X is LESS THAN 8. It cannot be equal to 8, because the inequality says that it is less than 8.

      Or if we were to see X > 8, we would know that whatever was in X was more than 8. That's important because the inequality says that whatever X has will be more than 8, so to make this inequality "true", so to speak, X has to be greater than 8. If we stick 7 into it, for instance, we would end up with 7 > 8, which of course is FALSE.

      What about X >= 8, though? (If you see this in textbooks, you'll actually see a little more-than symbol with a line under it instead of a more than sign and an equals sign. It is the same thing, however. )

      Anyway, >= is basically the same thing as the > symbol, except that in the > symbol, only numbers that are GREATER will make it true, but in the >= symbol, numbers that are greater will make it true, but numbers that are equal will ALSO make it true, which is why we also see the equals sign. It is the more-than-or-equal-to sign.

      Is 8 >= 8 true? Well, 8 is equal to 8, so yes, this is true, since the >= will have the inequality true when both sides are equal, or when the left side is more than the right side. In either of these cases, the inequality is true, so this is TRUE.

      Is 10>=8 true? Well, 10 is more than 8, so yes it is!

      7 >= 8 true? No, it is not, since 7 is less than 8, which will not make this true. So this is FALSE.

      Or what about the <= symbol? Well, this is the same thing, except only numbers that are less than, or equal, will make it true. It is the less-than-or-equal-to symbol. (Again, in textbooks, you'll see this symbol as a normal less-than symbol, but with a little line under it. The <= thing is used on computers to make it easier to type, so that's why you see it like that. )

      is 9 <= 8 true? No, it is not, since 9 is more than 8.

      Is 7 <= 8 true? Yes, 7 is less than 8, so this is true.

      is 8 <= 8 true? Well, since 8 is equal to 8, this is true.

      I hope that this helps! I know this isn't really all that detailed as how to solve them, but I hope that this is still helpful in at least understanding how they relate to equations.
      (115 votes)
  • aqualine ultimate style avatar for user LOLMAN
    at it says multiply both by 2/3 cant you just divide both by 3 and then multiply them by 2?
    (25 votes)
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  • winston baby style avatar for user yanicknorman
    So if it says that Old Maple Farms has EXACTLY 1000 more apples than River Orchards
    does this whole thing become an equality statement? like;
    M = R + 1000
    M/3 = (R + 1000)/3
    M/3 + x = (R + 1000)/3 + x
    or am i missing something?
    (21 votes)
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  • male robot johnny style avatar for user kehinde awofeso
    what is the difference between equation and inequalities
    (12 votes)
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  • duskpin ultimate style avatar for user Mark
    I think i understood but where in life will this be usable?
    (13 votes)
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  • blobby green style avatar for user Elbert,Roen
    who here was forced to do this.
    (16 votes)
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  • blobby green style avatar for user mssunshinelady2003
    I'm just wondering why the A isn't deducted from the equation since it's the same on both sides. Thereby leaving the original 2/3M > 2/3R. I could understand the extra steps if the increases were different, but this seems like extra steps for the sake of extra steps. Mind you, I am refreshing after 30+ years out of Alg 1 &2.
    (6 votes)
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  • blobby green style avatar for user steven foster
    I'm failing math... I'm in seventh-grade and my teacher doesn't teach... Help!
    (6 votes)
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  • orange juice squid orange style avatar for user Zahi Farah
    At s, Sal says that "going down by a 1/3 is the same as multiplying by 2/3". If I want to verify this, how can I do that? I seem to have a problem understanding ratios, it's really frustrating (especially with fractions).
    (6 votes)
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    • leaf red style avatar for user Grant Auleciems
      I don't see how that's possible either, except for the number 1... He must not be talking about regular numbers. Maybe he meant X minus one-third X = two thirds X... I checked it out and yes that does work. Maybe he made a mistake in his video. But I'm pretty sure he mean X minus one-third X = two thirds X.

      If you have anymore questions, ask me. :D
      (6 votes)
  • mr pants teal style avatar for user ultrabaymax
    Can an inequality have no answer?
    (6 votes)
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Video transcript

We're told that for the past few months, Old Maple Farms has grown about 1,000 more apples than their chief rival in the region, River Orchards. Due to cold weather this year, the harvests at both farms were down by about a third. However, both farms made up for some of the shortfall by purchasing equal quantities of apples from farms in neighboring states. What can you say about the number of apples available at each farm? Does one farm have more than the other, or do they have the same amount? How do I know? So let's define some variables here. Let's let M be equal to number of apples at Maple Farms. And then who's the other guy? River Orchards. So let's let R be equal to the number of apples at River Orchards. So this first sentence, they say-- let me do this in a different color-- they say for the past few years, Old Maple Farms has grown about 1,000 more apples than their chief rival in the region, River Orchards. So we could say, hey, Maple is approximately Old River, or M is approximately River plus 1,000. Or since we don't know the exact amount-- it says it's about 1,000 more, so we don't know it's exactly 1,000 more-- we can just say that in a normal year, Old Maple Farms, which we denote by M, has a larger amount of apples than River Orchard. So in a normal year, M is greater than R, right? It has about 1,000 more apples than Old Maple Farms. Now, they say due to cold weather this year-- so let's talk about this year now-- the harvests at both farms were down about a third. So this isn't a normal year. Let's talk about what's going to happen this year. In this year, each of these characters are going to be down by 1/3. Now if I go down by 1/3, that's the same thing as being 2/3 of what I was before. Let me do an example. If I'm at x, and I take away 1/3x, I'm left with 2/3x. So going down by 1/3 is the same thing as multiplying the quantity by 2/3. So if we multiply each of these quantities by 2/3, we can still hold this inequality, because we're doing the same thing to both sides of this inequality, and we're multiplying by a positive number. If we were multiplying by a negative number, we would have to swap the inequality. So we can multiply both sides of this by 2/3. So 2/3 of M is still going to be greater than 2/3 of R. And you could even draw that in a number line if you like. Let's do this in a number line. This all might be a little intuitive for you, and if it is, I apologize, but if it's not, it never hurts. So that's 0 on our number line. So in a normal year, M is has 1,000 more than R. So in a normal year, M might be over here and maybe R is over here. I don't know, let's say R is over there. Now, if we take 2/3 of M, that's going to stick us some place around, oh, I don't know, 2/3 is right about there. So this is M-- let me write this-- this is 2/3 M. And what's 2/3 of R going to be? Well, if you take 2/3 of this, you get to right about there, that is 2/3R. So you can see, 2/3R is still less than 2/3M, or 2/3M is greater than 2/3R. Now, they say both farms made up for some of the shortfall by purchasing equal quantities of apples from farms in neighboring states. So let's let a be equal to the quantity of apples both purchased. So they're telling us that they both purchased the same amount. So we could add a to both sides of this equation and it will not change the inequality. As long as you add or subtract the same value to both sides, it will not change the inequality. So if you add a to both sides, you have a plus 2/3M is a greater than 2/3R plus a. This is the amount that Old Maple Farms has after purchasing the apples, and this is the amount that River Orchards has. So after everything is said and done, Old Maple Farms still has more apples, and you can see that here. Maple Farms, a normal year, this year they only had 2/3 of the production, but then they purchased a apples. So let's say a is about, let's say that a is that many apples, so they got back to their normal amount. So let's say they got back to their normal amount. So that's how many apples they purchased, so he got back to M. Now, if R, if River Orchards also purchased a apples, that same distance, a, if you go along here gets you to right about over there. So once again, this is-- let me do it a little bit different, because I don't like it overlapping, so let me do it like this. So let's say this guy, M-- I keep forgetting their names-- Old Maple Farms purchases a apples, gets them that far. So that's a apples. But River Orchards also purchases a apples, so let's add that same amount. I'm just going to copy and paste it so it's the exact same amount. So River Orchards also purchases a, so it also purchases that same amount. So when all is said and done, River Orchards is going to have this many apples in the year that they had less production but they went and purchased it. So this, right here, is-- this value right here is 2/3R plus a. That's what River Orchards has. And then Old Maple Farms has this value right here, which is 2/3M plus a. Everything said and done, Old Maple Farms still has more apples.