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Compound Inequalities

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Today we're going to talk about compound inequalities. A compound inequality is made up of two or more inequality symbols. So for example, 3 is less than x is less than 5. So we're going to look at two different types, and compound inequalities and or compound inequalities. And we'll do some examples of each.

So let's start by looking at and compound inequalities. An and compound inequality, such as x is less than 5 and x is greater than 0, is a set of two or more inequalities. And your solution set has to satisfy both or all of the inequalities. So clearly this is an and compound inequality because of the word and. And if we picked a value, like x equals 3, we know that would satisfy, that that would be in our solution set because 3 is less than 5 and 3 is greater than 0.

Another way to write an and compound inequality is by combining them together and writing that x is less than 5 and x is greater than 0 like this. So we could graph this on our number line by using open circles at 0 and 5, because they cannot be exactly equal to 0 or 5, and then showing that x can be the range in between but not including 0 and 5.

We could also write this in set notation, where we have the solution set is all x values such that 0 is less than x, which is less than 5. And we could write this in interval notation, where 0 is our lower bound of our solution set, 5 is our upper bound of the solution set, and we use curvy brackets or parentheses because it again cannot include these two numbers.

So let's do an example of or compound inequalities. So here's an example of an or compound inequality. x is less than or equal to negative 1 or x is greater than or equal to 2. So clearly this is an or compound inequality because of the word or. And an or compound inequality is still a set of two or more inequalities, but the solution set does not need to satisfy all of the inequalities, just any of them.

So for example, x equals 4 would be in our solution set for this compound inequality, because it satisfies the x is greater than or equal to 2 inequality. So let's see what this would look like plotted on our number line. I've got closed circles above my negative 1 and my positive 2, because those exact values are in my solution set and I want to show that x can be also greater than 2 and x can be less than negative 1.

So if I also wanted to write this in interval notation, we would have that we're starting at negative infinity and we're going up to negative 1. We are including negative 1 in our solution set, so we have a square bracket. And we always use a parentheses around positive or negative infinity.

And in our second inequality or our second interval, our first value includes 2, so square bracket. And we're going all the way up to positive infinity, again, using a parentheses with infinity. And then we use a U to show that this is the union between these two individual intervals. If we want to write this now in set notation, we want to say that the set includes all x values such that x is less than or equal to negative 1 or x is greater than or equal to positive 2.

So let's do another example of an and compound inequality. So I'm going to start solving this as I would an equation. I'm going to subtract 6 to cancel out the adding 6. And then I need to do the same thing on the other side of both inequalities. So this is going to become 3x is less than or equal to 12, which is greater than or equal to 0.

Then I'm going to divide by 3 again on the other side of both inequalities. My 3's will cancel out. And I now have that 0 is less than or equal to x, which is less than or equal to 4.

So if I want to plot this on my number line, I'm going to start with a closed circle at 0 and a closed circle at 4 and show that x can be anything in between those. If I want to write this in set notation, I'm going to show that x can be all x values such that 0 is less than or equal to x, which is less than or equal to 4. And if I want to write this in interval notation, I'm going to use my square brackets, starting with my 0 and going up to my 4.

All right, another example of an or compound inequality-- I'm going to solve both inequality separately. So I'm going to start here by subtracting 2 on both sides to cancel this out, leaving me with 4x is less than 8. Then I'm going to divide by 4 on both sides. This will cancel. S now I have x is less than 2 for my first inequality.

For this one, I'm going to start by subtracting 6. This will cancel. And I've got 3x is greater than 9. And I'm going to divide by 3 on both sides. My 3's will cancel. And I've got x is greater than 3. So my solution set is x is less than 2 or x is greater than 3.

So plotting this on my number line, I've got an open circle at 2. And x can be anything less than that, but again, not including 2. And x can be greater than 3, but not equal to, so we'll use our open circle and then show everything to the right.

Now, if I want to write this in set notation, I'll have x can be all values such that x is less than 2 or x is greater than 3. And in interval notation, we will have-- let's see-- a curved bracket, because we've got negative infinity. And x can be up to but not including 2-- so another curved bracket or parentheses. And our second interval goes from 3, but not including-- so curved bracket-- all the way up to infinity. And we want to show that our solution set is the union between these two individuals sets.

All right, let's go over our key points from today. And compound inequalities have two or more inequalities. And the solution set has to satisfy all of the inequalities. Or compound inequalities also have two or more inequalities. But the solution set satisfies any but not necessarily all of the inequalities. And then when you're solving an and compound inequality with an algebraic expression between two inequality symbols, any operation done between the symbols must also be done on the other side of both inequality symbols.

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