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