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## Algebra basics

# 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.

## Want to join the conversation?

- I still don't get what an inequality is too well. I'm sorry, but can someone explain?(20 votes)
- 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.(83 votes)

- at2:51it says multiply both by 2/3 cant you just divide both by 3 and then multiply them by 2?(19 votes)
- Yes you can but that's the same thing as multiplying both sides by 2/3.

Because 2*1/3 = 2 divided 3 = 2/3(17 votes)

- 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?(16 votes)- You are correct, it becomes an equality statement. And no, you're not missing something.(8 votes)

- what is the difference between equation and inequalities(8 votes)
- An equation has an equal sign, inequalities have a less than, greater than, less then or equal to, or greater than or equal to sign.(11 votes)

- who here was forced to do this.(11 votes)
- I think i understood but where in life will this be usable?(5 votes)
- An individual must know quite a bit of math to be a, Doctor , Chemical Engineer , Electronic Engineer , M B A , Astronomist , Physicist , Geologist , Astonaut , Economist , Cosmologist , etc.(5 votes)

- 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.(4 votes)
- Having it as M > R also serves the same purpose because 2/3 didn't change the inequality either.

But i think its to show the steps and how we actually reach to that conclusion by working it out.(7 votes)

- I'm failing math... I'm in seventh-grade and my teacher doesn't teach... Help!(4 votes)
- What do you need help on? This is very urgent, I really want to help you.(4 votes)

- At2:29s, 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).(4 votes)
- 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(4 votes)

- Can an inequality have no answer?(4 votes)
- Yes, it can! For example, 1+x<x wouldn't make sense.(3 votes)

## 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.