Video Transcript: Two-Step Inequalities

Two-step inequalities 

We have the inequality 2/3 is greater than negative 4y minus 8 and 1/3. Now, the first thing I  want to do here, just because mixed numbers bother me-- they're actually hard to deal with  mathematically. They're easy to think about-- oh, it's a little bit more than 8. Let's convert this  to an improper fraction. So 8 and 1/3 is equal to-- the denominator's going to be 3. 3 times 8 is  24, plus 1 is 25. So this thing over here is the same thing as 25 over 3. Let me just rewrite the  whole thing. So it's 2/3 is greater than negative 4y minus 25 over 3. Now, the next thing I want  to do, just because dealing with fractions are a bit of a pain, is multiply both sides of this  inequality by some quantity that'll eliminate the fractions. And the easiest one I can think of is  multiply both sides by 3. That'll get rid of the 3's in the denominator. So let's multiply both sides of this equation by 3. That's the left-hand side. And then I'm going to multiply the right-hand  side. 3, I'll put it in parentheses like that. Well, one point that I want to point out is that I did not  have to swap the inequality sign, because I multiplied both sides by a positive number. If the 3  was a negative number, if I multiplied both sides by negative 3, or negative 1, or negative  whatever, I would have had to swap the inequality sign. Anyway, let's simplify this. So the left hand side, we have 3 times 2/3, which is just 2. 2 is greater than. And then we can distribute this  3. 3 times negative 4y is negative 12y. And then 3 times negative 25 over 3 is just negative 25.  Now, we want to get all of our constant terms on one side of the inequality and all of our  variable terms-- the only variable here is y on the other side-- the y is already sitting here, so  let's just get this 25 on the other side of the inequality. And we can do that by adding 25 to both  sides of this equation. So let's add 25 to both sides of this equation. And with the left-hand side, 2 plus 25 five is 27 and we're going to get 27 is greater than. The right-hand side of the  inequality is negative 12y. And then negative 25 plus 25, those cancel out, that was the whole  point, so we're left with 27 is greater than negative 12y. Now, to isolate the y, you can either  multiply both sides by negative 1/12 or you could say let's just divide both sides by negative 12.  Now, because I'm multiplying or dividing by a negative number here, I'm going to need to swap the inequality. So let me write this. If I divide both sides of this equation by negative 12, then it  becomes 27 over negative 12 is less than-- I'm swapping the inequality, let me do this in a  different color-- is less than negative 12y over negative 12. Notice, when I divide both sides of  the inequality by a negative number, I swap the inequality, the greater than becomes a less  than. When it was positive, I didn't have to swap it. So 27 divided by negative 12, well, they're  both divisible by 3. So we're going to get, if we divide the numerator and the denominator by 3, we get negative 9 over 4 is less than-- these cancel out-- y. So y is greater than negative 9/4, or  negative 9/4 is less than y. And if you wanted to write that-- just let me write this-- our answer is y is greater than negative 9/4. I just swapped the order, you could say negative 9/4 is less than  y. Or if you want to visualize that a little bit better, 9/4 is 2 and 1/4, so we could also say y is  greater than negative 2 and 1/4 if we want to put it as a mixed number. And if we wanted to  graph it on the number line-- let me draw a number line right here, a real simple one. Maybe  this is 0. Negative 2 is right over, let's say negative 1, negative 2, then say negative 3 is right  there. Negative 2 and 1/4 is going to be right here, and it's greater than, so we're not going to  include that in the solution set. So we're going to make an open circle right there. And  everything larger than that is a valid y, is a y that will satisfy the inequality.

Two-step inequality word problem: apples 

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. 

Two-step inequality word problem: R&B 

A popular R&B band recently returned from a successful three-city tour where they played to at least 120,000 people. My brain immediately says that's greater than or equal to 120,000. If they  had an audience of 45,000 in Mesa and another 33,000 in Denver, how many people attended  their show in Las Vegas? So let's say Las Vegas, I'll just use l for Las Vegas. So the number of  people who attended their show in Las Vegas plus the number that attended their show in  Mesa, which is 45,000, plus the number of people that attended their show in Denver, which is  33,000-- those are three cities right there, Las Vegas, Mesa, and Denver-- that has to be at least  120,000 people. Or another way of interpreting that is greater than or equal to 120,000. So to  figure out how many people attended their show in Las Vegas, we just solve for l on this  inequalty. So if we simplify this left-hand side, we get the number of people in Las Vegas plus--  what's 45,000 plus 33,000, that is 78,000-- 78,000 is going to be greater than or equal to  120,000. Now to isolate the l on the left-hand side of the inequality, we can subtract 78,000  from both sides. So minus 78,000, minus 78,000 on the left-hand side, these cancel out. And  we're just left with the number of people who attended the show in Las Vegas is going to be  greater than or equal to 120,000 minus 78,000. So 120,000 minus 80,000 is 40,000, and it's  going to be another 2,000. So the number of people who attended Las Vegas is going to be  greater than or equal to 42,000 people. And we're done, that's it