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Hi, I’m Rob. Welcome to Math Antics.
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In a previous video, we learned that the number line is a helpful tool
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for showing how numbers relate to each other.
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It’s organized so that numbers increase in value as you move from left to right,
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and they decrease in value as you move from right to left.
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In other words, some numbers are greater than other numbers
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and the number line provides a quick way to compare them.
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In math, those sorts of comparisons are called “inequalities”.
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Inequality?!
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Why… that’s unconstitutional!
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Oh, hey there.
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But... don’t worry… I’m talking about numbers being inequal… not people!
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Ya know… cuz this is a math video.
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Ah… I see.
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Well, this must be the wrong program.
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I’m… I’m really more of a history guy actually.
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Well, pardon the interruption.
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No worries.
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Anyway, to help you understand what inequalities are,
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let’s first start with a different kind of mathematical comparison called “equations”.
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You’ve heard of equations, right?
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Equations use a special symbol called “the equal sign”,
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…like 1 + 1 = 2.
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This simple equation tells you that if you have 1 and add 1 more to it,
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the value you get exactly equals 2.
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So equations tell you when two things have the same value, which means they’re equal.
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Inequalities on the other hand, tell you when things are not equal.
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They use two different symbols called the “greater than sign” and the “less than sign”.
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These signs each have a bigger, open end that always faces the bigger value,
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and a smaller, pointed end that always points to the smaller value.
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My teacher used to tell me to imagine that the signs were alligator mouths
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that always wanted to eat the biggest number!
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But you may be wondering,
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“Why do inequalities get two different symbols when equations only get one?”
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Well, it has to do with the order that we write things in.
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…take our simple equation for example.
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If we switch the order so it reads, 2 = 1 + 1, that’s still true, right?
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The order didn’t make any difference.
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But what about the inequality 5 > 3?
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That’s true because 5 is more than 3,
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but what if we switch the order of the numbers like this?
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Uh oh… if we do that, we don’t have a true statement anymore.
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We’re still using the ‘greater than sign’ and 3 is not greater than 5.
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Notice the open part of the symbol is facing the smaller number, which is wrong.
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If we want the statement to still be true when we switch the order of the numbers,
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we also have to switch the symbol to the ‘less than sign’ so it reads 3 is less than 5.
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But did you notice that the ‘less than sign’
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looks just like the ‘greater than sign’ facing the other direction.
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That means you can think of the ‘greater than’ and ‘less than’ signs as just one symbol
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that’s read differently depending on which direction it’s facing when you read it.
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If you read it starting from the bigger, open end,
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you say “is greater than” as you go past it.
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But if you read it starting from the smaller, pointed end,
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then you say “is less than” as you go past.
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The benefit of thinking about the symbols this way (like just one symbol)
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is that no matter which way it’s facing,
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you won’t get confused about the relationship of the things that it’s comparing.
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Oh… and before we move on, I want to show you two other inequality symbols
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that can be made by combining the ‘greater than’ and ‘less than’ signs with the equal sign,
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but I’m afraid you won’t be very impressed by their names.
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This one is called the “greater than OR equal to sign”
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and this one is called the “less than OR equal to sign”.
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You’ll see why these combined symbols are needed in just a minute.
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For now we’re going to move on and see how basic inequalities look
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when they are graphed on the number line.
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Let’s start out with a number line and a very simple mathematical statement: n = 3
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Ah, I know what you’re thinking;
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That’s got an equal sign, so it’s an equation, not an inequality.
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That’s a good observation!
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And you may also be wondering, what does the letter ’n’ mean?
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Why is there a letter instead of a number?
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Well, letters are used all the time in math as you’ll learn when you get to basic algebra.
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They can represent numbers that aren’t know yet,
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or values that can change,
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or even groups of numbers.
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In this case, just think of ’n’ like a placeholder for a number that’s a possible answer.
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Like, n = 3 just means “a number that equals 3”.
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Can you think of a number that equals 3?
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Yep, it’s pretty obvious that the answer is 3.
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And graphing that on the number line is easy. We just draw a point at 3.
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But now, what if we change this equation into an inequality using the greater than sign?
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That would mean “a number that’s greater than 3”.
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Can you think of an answer for that?
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Well, 4 would definitely work.
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If we substitute 4 in place of ’n’, we’d get the inequality 4 > 3 which is a true statement.
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But there are lots of other numbers that would work also.
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For example, we could have picked 5 instead,
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because if we substitute 5 in place of ’n’ we’d get the inequality 5 > 3 which is also true.
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So inequalities can have multiple answers.
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In fact, any number located to the right of 3 on the number line would make this inequality true
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because values increase as you move to the right.
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6 would work.
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10 would work.
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In-between numbers like 7.5 would work.
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No matter which number we choose from the right side of the 3,
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it will be a valid answer to the inequality n > 3
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And notice, the more answers we add to the graph,
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the more they’re starting to look just like a solid line.
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That’s how you graph a simple inequality like this.
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Since there are an infinite number of possible answers,
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instead of drawing an infinite number of points,
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you just draw a line to cover all of those possible answers at the same time.
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We even put an arrow at the end of that new line
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to show that it goes on forever just like the number line itself.
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Cool, so the graph of the equation n = 3 is a single point since there’s only one valid answer.
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But the graph of the inequality n > 3 is a line since there are infinite possible answers
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as long as they are all to the right of 3.
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Nothing to the left of 3 is a valid answer because those values are all less than 3.
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But what about the point located exactly at 3?
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Would the be a valid answer to this inequality?
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Well no because if we substitute that in, we’d get the statement 3 > 3 which isn’t true.
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Fortunately, mathematicians came up with a clever way to show this on our graph.
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Instead of putting a solid dot at 3 like you would if it was a valid answer,
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you put a small hollow dot (or a circle)
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to show that the value 3 itself is not included in the set of possible answers.
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Just imagine that the open dot means that value is missing from the set of answers.
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Ah, but remember those other inequality symbols I showed you?
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…the ones that are combined with the equal sign?
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If we had the inequality n ≥ 3, the graph would look almost the same
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except that we’d fill in the point at 3 to show that 3 is included in the set of possible answers.
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In other words, if you want to specifically include the value exactly at the boundary of the inequality,
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you can use one of the combined signs to include it. …so it doesn’t feel left out.
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Think of it like this…
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Suppose you’re 10 years old at an amusement park,
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and a sign says that you have to be “age 10 or older” to ride a particular ride.
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That could be shown as the inequality Age ≥ 10.
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And you’d be in luck because your age (10) is included in the set of ages that are allowed to go on the ride.
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But if the sign said “you must be older than 10”. That would be Age > 10.
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And you wouldn’t be able to ride because your exact age is not included,
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and you’d feel sad
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…and angry
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…and so disappointed
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…and a little jealous at your older sister who did get to ride
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…and you wouldn’t even be happy with your ice cream anymore
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…and you’d want your money back
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…and you’d want to just go home by yourself.
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Okay, let’s try another example to make sure you’ve got the idea.
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Let’s graph the inequality n < 7.
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Just like with the last example,
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you can probably think of many values of ’n’ that would make this statement true.
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6 is less than 7.
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So is 4.
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So is 1.
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So is 0.
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Any number that’s to the left of 7 on the number line will work for this inequality
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because values decrease as you go from right to left on the number line.
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So again, since there are an infinite number of possible answers,
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we’ll just draw a line to cover all of them.
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And as before, we’ll leave an open dot exactly at 7 to show that point is not included
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since 7 is not less than 7.
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Of course, if we had the inequality n ≤ 7 instead,
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we’d fill in that point to include 7 in the set of valid answers.
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Oh… and some of you who are a little ahead in math may realize that
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there are other numbers that meet the requirement of being less than 7
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that are currently not shown on this graph because they are found to the left of zero on the number line.
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But don’t worry about those negative numbers for now.
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That’s a more advanced topic that we cover in a future video.
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Making sense so far?
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Good.
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Let’s move on to something just a bit more interesting.
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What would happen if we did a little re-arranging and combined the two inequalities we just graphed like this?
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We end up with something new called a double or compound inequality.
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But what does it mean?
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Well, this new inequality is defining a relationship between ’n’ and two different numbers,
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instead of just one like before.
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If you read it from left to right you’d say, “3 is less than ’n’ which is less than 7”.
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That’s correct, but it’ not as clear as it could be.
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Instead, what if we read this starting from the middle and then read in both direction (one at a time)?
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Reading to the left we’d say “n is greater than 3”
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because even though this is the ‘less than sign’, we started reading from the bigger open end facing the ’n’.
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Then, starting from the middle but reading to the right this time, we’d say “and less than 7”.
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In other words, it’s asking us what numbers are in-between 3 and 7.
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You can probably think of a lot of numbers that would fit that description.
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…like the number 5. 5 is in-between 3 and 7.
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Like before, we want our graph to include all of the valid answers for this inequality.
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Since any number that’s in-between 3 and 7 will work, we’ll draw a line from 3 to 7 like this.
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Then we’ll put a hollow dot at each end of that line to show that those numbers aren’t included.
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We got this inequality by combining the first two inequalities we graphed, right?
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Well, notice that if we put those graphs right next to each other,
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the section where they overlap is the same as the answer set of our new compound inequality.
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Pretty cool, huh?
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And one of the really useful things about compound inequalities is that
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they’re great for specifying a range of values. …like a price range.
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Suppose you want to buy a bicycle. How would you use this idea
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to specify that you want a bike that’s greater than $50 but less than $200?
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Well, using P to stand for the price,
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you could write those conditions individually as P > 50 and P < 200.
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But you could also combine them to get this, which means that P is greater than 50 AND less than 200.
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And of course, if you were okay with getting a bike that was exactly $50,
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you could change this first symbol to include the equal sign,
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and if you were okay with it costing exactly $200
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you could change the second symbol as well.
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And what about a temperature range?
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How would you specify that your cat prefers that you to keep the thermostat set
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between 68 and 72 degrees Fahrenheit? [meow]
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Well, using T for temperature you could say that T > 68 and T < 72.
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Combining them into one statement gives you this compound inequality:
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…The purrrfect temperature range for your furry friend. [cat purring]
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Alright, as you can see, inequalities are really useful in math and in everyday life.
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They help you compare numbers to show which has the greater value
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and they help you specify conditions or ranges of acceptable values.
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Yeah, I guess they aren’t as bad as I thought at first.
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Are you still here? I thought you had another show to do.
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Ah yeah… Well, it turns out that show got cancelled.
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I’m just here waiting for my Uber to pick me up.
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Okay well hopefully they’ll be here soon cuz I’m kinda trying to finish this video.
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As you’ve heard me say before, the best way to learn math is to practice…
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And the best way to learn history is to read a lot about it!
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Alright… As always, thanks for watching Math Antics, and I’ll see ya next time!
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Uh, not me ya won’t… my ride is here!
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Learn more at www.mathantics.com
