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Introduction to Inequalities

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SPEAKER: Today we're going to talk about inequalities. An inequality is just a mathematical statement that says that two quantities are un-equal in value. So for example, here I've got three is less than five. So this statement is saying that three has a smaller value than five. And we use this inequality symbol, the less-than symbol, to show that three is smaller than five. Notice that the inequality symbol is opening up towards the number with the bigger value. Another example would be negative two is greater than negative seven. So this statement is saying that negative two has a greater value. We use this symbol, the greater-than inequality symbol, to show that negative two has a greater value. And again notice that the inequality symbol is opening up towards that quantity with the bigger value. So let's do some examples of using inequalities with algebraic expressions.

So we can use inequalities with algebraic expressions. For example, I can have x is less than three, which means that x can be any number that is less than three and that would make this mathematical statement true. Similarly here, x is greater than negative two means that x can be any number that is greater than negative two and satisfy this inequality.

We also have a symbol called "less-than or equal to." Here, x can be any number that is less than five but also equal to five. And that would satisfy this inequality. And similarly, x is greater than or equal to zero means that x can be any number that is greater than zero but could also equal exactly zero.

So let's do some examples graphing inequalities on number lines. My first example says that x is less than three. So I know that x cannot be exactly equal to three but can be anything smaller. So I'm first going to show that x cannot be exactly equal to three. And we do that with an open circle on our number line right above the three. So an open circle means that your value for x cannot be exactly equal to the value that it's on top of. I also want to show that x can be less than three. So I'm going to use an arrow pointing to the left, or pointing to the numbers that are less than three. And now I see that this region represents all the solutions to my inequality, x is less than 3, and then any value in this highlighted region is going to satisfy my inequality.

For my second example, I've got x is greater than or equal to negative one. So first I'm going to show that x can be exactly equal to negative one. And I do that by using a closed circle on my number line right above negative one. And then I want to show that x can also be anything greater than negative one. So I'm going to use an arrow pointing to the right, towards the numbers that are greater than negative one. And now this represents all the solutions to my inequality, x is greater than or equal to negative one. And so any value of x in this highlighted region, including negative one, is going to satisfy my inequality.

For my third example I've got that x is going to be greater than negative three but less than or equal to two. So my values for x that are going to satisfy this inequality have to be bigger than negative three and less than or equal to two. So I'm going to start by putting an open circle above my negative three, because I see that x cannot be exactly equal to negative three. And then I'm going to use the closed circle above my two because I know that x can be equal exactly to two.

And then I want to show that x is greater than negative three-- so that's going to be numbers in this direction-- but that x is also smaller than or equal to two-- so that will be numbers in this direction. So I can see that my solution is going to be in between my two points. So using this highlighted region I can see that any value that's in this region is going to satisfy find my inequality. So again, any number greater than negative three or less than or equal to two will satisfy this inequality.

So for my last three examples I'm going to start with a graph of an inequality. And I'm going to write the inequality that is associated with the shaded region on the number line. So first I've got a graph that starts at one. But because of the open circle I know that the solution set represented by this inequality is not going to exactly include one. But it is going to be including numbers that are smaller than one because it's shading over numbers that are smaller than one. So I will start with my x. I know that it is less than but not exactly including the number of one. So this inequality represents the shaded region on this graph.

Here I have a closed circle at zero and then I have numbers shaded that are greater than zero, they're going to the right. So I'll start again with my variable x. I know that the solution set includes exactly zero and then it also includes numbers that are greater. So greater or equal to zero.

Then I have the shaded region that is in between two values. So I see I have a open circle at negative four, which tells me that is the solution set does not exactly include negative four. But it does include numbers that are greater than negative four, all the way up to and including the number two. And I know that it includes two again because of the closed circle. So I have x in the middle. I have numbers that are greater than but not exactly equal to negative four and numbers that are less than or equal to the value of two.

So let's go over our key points from today. We use inequalities to show that two quantities are not equal in value. We can use a number line to show a range of values that can satisfy an inequality expression. And when we're plotting an inequality on a number line we use an open circle for the symbols less-than and greater-than and use a filled in circle for the symbols "less-than or equal to" and "greater-than or equal to." So make sure you get these points in your notes, if you don't have them already, so you can refer to them later.

So I hope that these key points and examples helped you understand a little bit more about inequalities. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.