WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at solving linear inequalities in one variable.
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To solve linear qualities, you will see that the process is not so bad.
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But we are willing to take a little bit of time just to highlight the difference between inequalities and equations,
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after looking at the difference between the two and that will cause a little bit of a problem.
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You have to be careful on how we describe our solutions for inequalities.
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Then we will get into a bunch of different examples on how we can finally solve these linear inequalities.
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Let us get started.
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When you are looking at an inequality versus an equation and you try to figure out what is the difference.
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One of the main differences, of course you will see a different symbol in there, one of our inequality symbols.
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It is these guys that we saw earlier.
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We have our greater than symbol, greater than or equal to symbol, our less than symbol or less than or equal to symbol.
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It is a little bit more than just having an additional symbol in there that makes this different.
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What makes an inequality different from an equation has to do with the solutions that you get out of them.
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To highlight this I will go over some quick examples of equations and get to one of those inequalities.
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Consider this first one here, 3x = 15.
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If you are going to solve that, it would not take probably that long.
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Just simply divide both sides by 3 and you get that x = 5.
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Now what that says the only number, the only one out of all possible numbers that will make that equation true is just 5.
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It is like this little isolated point out on the number line.
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Some equations might actually have more than one solution, for example, this next one x² = 9.
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If you are going to look for numbers that will make that equation true, you will actually find two of them.
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When x = 3 and you square it you will get 9, but also when x = -3 and you square it you still get 9.
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You know it is a little different that you have two solutions, but both of the solutions those are the only two things that will make it work.
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It is like two little isolated points on your number line.
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To contrast that to this first inequality over here, 4x is greater than or equal to 8.
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If you try to figure out what numbers make that true, you would end up with a huge, huge list of numbers that work.
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Let us start picking out some examples.
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Also, what if I said that x was equal to 2.
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If you substitute that in, you would have 4 × 2 is greater than or equal to 8,
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which would simplify to 8 is greater than or equal to 8, which is, of course true.
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But it is not the only thing that works out.
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You could also put in 3, 4, you can get a little creative and put in some other ones like 4 1/2, and all of these would be solutions.
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Trying to list out solutions for an inequality is simply not the way to go.
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The reason for that is if you look at the numbers that would make an inequality true, you will get an entire range of values.
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So, not just in a little isolated point, they are in whole ranges of values on our number line.
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Since we are after a whole range of values other than these isolated points, we are going to have to be a little bit careful on how we describe those.
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In fact, there are many ways that you can describe the solution to an inequality.
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Some you seen before and some might be a little new, let us go over them.
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The first way that you can describe the solution is to simply use our inequality symbols that will be like this guy down here.
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Maybe I’m trying to describe all the numbers that are less than 4, I can simply write x is less than 4.
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We can also use our interval notation.
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This is a good way to mimic the number line and that it gives us the lowest number or the starting point and the endpoint.
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It also gives us information on whether we should include the number or not.
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Since this one has parentheses, I know that the 4 is not included.
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It is simply a different way of writing the same information.
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This interval says I'm looking at all the values that are less than 4.
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We could also describe our solutions a little bit more visually using a number line.
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The way we do this is we have a number line and we shade in the solution.
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To indicate whether the endpoint is included or not, you can use an open circle or close circle.
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The open circle here means it is not included, so I would have used a close circle if I had x is less than or equal to 4.
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And one last way that you might see, I do see this in a few Math books is known as set builder notation.
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It looks a little clunky and that we have a couple of curly brackets thrown in there
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but let me show you how you can interpret the set builder notation.
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First of all, we identify some sort of variable at the very beginning.
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We are looking for all x values.
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And after that you will notice that we borrow our inequality.
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This is the rule of all of the numbers that we will include.
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That little line you see in between, we say that is L and it is just the divider or it separates both those sections.
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The way you would read the set builder notation is the set of all x’s such that x is less than 4.
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We know that we are building for x values and we know which x values would get in there.
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I’m not going to focus a whole lot on that set builder notation but you will definitely see me use some inequalities, some number lines
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and some interval notation just to describe our solutions.
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Now that we know little bit about what makes an inequality different, we know little bit more on how to describe the solutions,
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let us get into how we can actually start solving these things.
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The good news is when it comes to solving linear inequality, we use the same techniques as we do with solving a normal linear equation.
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There is one additional rule that you want to be aware of.
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When you multiply or divide by a negative number then you want to flip the inequality sign.
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Now that rule applies to multiplying and dividing, so be careful not to try and use it if you are only adding and subtracting.
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Use it only for multiplying and dividing.
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I often get lots of questions like why is it that you have to flip the sign when your doing the multiplication and division by negative numbers.
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I think a quick example will help you understand why that is.
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Let us take a nice inequality like 2 < 3, so seems pretty simple.
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We know that 2 < 3 but watch what happens if I multiply both sides by a negative number.
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-2 and -3, now which one is less than the other one?
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If we take a peek at our number line, you will see that it is the -3 that is now the smaller number since it is on the left side of -2.
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This means that if we want to preserve the trueness of our statement, we also must flip that sign to compensate for that negative that we threw in there.
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It is easy to see with numbers like this but in a moment when we start dealing with x’s and unknowns,
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we still have to remember to flip the sign when we divide or multiply by that negative number.
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I think we have all the information we need.
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Let us go ahead and get into the solving process and see if we can tackle some of these inequalities.
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We want to solve the following inequality then write our answer using a number line and using interval notation.
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Okay, when I solve an equation usually I try and isolate the x's and I'm going to do the same thing here.
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I will do that by first moving the 5 to the other side, 3x < 11 - 2x and then we will go ahead and we will add 2x to both sides.
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5x < 11, let us go ahead and divide both sides by 5.
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I will get that x < 11/5.
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This could represent my solution using just the inequality symbol.
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Now I want to go and have a look at the number line and its interval notation.
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Here is a nice quick sketch of a number line.
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I'm looking to shade in all values that are less than 11/5.
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If I want to rewrite that, 5 goes into 11 twice with the remainder of one, so it is the same as 2 1/5.
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It is a little bit greater than 2, but not much.
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Notice how I’m using an open circle there because our inequality is strict.
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We do not want to include that value.
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I will shade in everything that is less than my 2 1/5.
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This number line represents all of the solutions to my inequality, I could use anything on that side and it worked.
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As long as we have our number line, let us go ahead and also represent our solution using interval notation.
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We want to think of where we start on the left side, it looks like we are starting way down at negative infinity.
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We go all the way up to 11/5 but I will use a parenthesis since we do not include that value.
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I have represented my solution now in 3 different ways. Not bad.
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Let us look at another, for this one we are looking to solve 13 - 7x = – 4.
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Let us start off by trying to get those x's isolated and we will do that by first getting them together.
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I'm going to move this 10x to the other side, so I'm subtracting here.
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I do not have to worry about flipping any signs nothing like that yet, 13 - 17x = -4.
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Let us go ahead and subtract 13 from both sides, we have -17x = -17.
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Be very careful in this next step.
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I need to get the x all by itself, in order to do that I will divide by -17.
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I will do all of the algebra normally -17 ÷ -17 = 1.
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I'm also going to remember what to do with that sign, we are going to flip it around.
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I have it x = 1.
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Let us get to our number line and go ahead and represent the solution.
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I have the number 1,we will make a nice solid circle since this is or equal to
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and we will go ahead and shade in everything less than that since it says x < 1.
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Now that I have my number line, let us represent this using our interval notation.
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On the left side we are starting at way down to negative infinity and going all the way up to 1.
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We want to include the one so we will use a bracket.
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Let us try one more example.
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In this last one I picked a little bit more of a word problem so you can see that inequalities are important even in applications.
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This one says that Joanna is hiring a painting company in order to get her house painted
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and the company has two plans to choose from, you can choose plan A.
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In plan A they charge you a flat fee of $250 and $10 for every hour that it takes to paint the house.
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For plan B, they do not charge a flat fee but they will charge $20 per hour.
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The question is when will plan B be more expensive than plan A?
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Pause for a moment and think why would we be using an inequality here?
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Why do not we just set up an equation?
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What I'm looking for is not one specific time, but all times when plan B will be more expensive.
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It makes a little bit more sense to go ahead and set up an inequality for this word problem.
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Let us go ahead and hunt down both of our situation.
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Let us have plan A and let us go ahead and have plan B.
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We need an unknown here, let us say x is the time to paint the house.
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Plan A cost a flat fee of $250 + $10/hour.
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$10/ hour is like our variable cost, that would be 10x, flat fee does not vary so +250.
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That expression will just keep track of the cost for plan A depending on how many hours it takes.
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Plan B has only a variable cost of $20/hour, I will just say 20x.
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Now comes the big question, when is plan B more expensive than plan A.
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B would be more expensive, I would use an inequality symbol of plan A < plan B
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and that is exactly how we will connect our expressions as well.
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10x + 250 < 20x and to completely answer this problem we just have to solve the inequality.
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Let us work on getting our x's together by subtracting 10x from both sides.
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250 < 10x, now we will divide both sides by 10.
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25 < x, now it is time to interpret exactly what we have here.
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x represents the time that it will take to paint the house.
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What I can see here is that anytime my time is more than 25 hours then I can be sure that plan B will be more expensive, there you have it.
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Also, be careful when setting up and solving these inequalities.
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Remember to go ahead and flip your inequality anytime you multiply or divide by negative number.
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