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Okay, we're talking about something called compound inequalities, which you should

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have

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learned last year, but we're going to kind of review a little bit.

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Let me explain a little more of this.

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Compound inequalities are two or more.

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We can have more than two, but typically it's just two inequalities.

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And then there's two different types.

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One's called an OR, one's called an AND compound inequality.

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For the AND compound inequality, a number is a solution if it's a solution to both

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inequalities.

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That's the AND.

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And the OR inequality is a number is a solution to the compound if it's a solution

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to one

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or the other inequality.

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So let me give you an example.

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So we have AND and OR inequalities.

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They will either say the words

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They will either use the word AND or the word OR. There is another way to write an

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AND inequality where you will not see the word AND.

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solution to the compound inequality the number must be a solution to this and

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and also a solution to this.

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So these are already solved for x.

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So all the solutions to x greater than or equal to 4

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Are all of these numbers correct?

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And solutions to this, x is less than 10, are all these solutions?

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So solutions to this compound inequality

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are just the numbers that are solutions to this and this.

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So what on the graph is going to be a solution?

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5.

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Whatever is in both green and blue.

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OK.

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So the numbers are in both green and blue would be is four in both colors?

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Yes. Yes and then everything here what about ten is ten in both colors? No. No

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because of the open circle there. Okay so this is the solution set to that

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compound and inequality.

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And so the solution set, of course that's the graph of the solution set.

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The actual solution set in interval notation is all the numbers between 4 and 10,

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including 4 but not including 10: [4, 10).

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If this compound inequality could be written in one line,

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Okay. If, uh, that's not the best way of writing it. Let's do it this way.

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That's this guy right here, right?

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And if I flip this guy around, same thing as 4 is less than

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or equal to x.

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So we can just put this guy right here.

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So if you ever see something like this,

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that's a compound inequality.

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4 is less than or equal to x, which is less than 10.

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Was it less than 10?

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OK.

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This is two inequalities in one, a two for one.

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Okay, if you ever see two inequalities in one,

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it's a compound and inequality.

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Has to be a solution to this and a solution to this.

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Where or inequalities have to be written separately,

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you have to use the word or in between.

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So x is greater than or equal to 5, or x is less than 0. You always see the word OR

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in between.

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word or in between

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OK, so in orange is the solution to the x is less than zero, green is solution to x

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greater than or equal to 5.

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So the solutions to the compound inequality

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is the ones that are in green or orange.

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We draw these.

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And then to write this in interval notation,

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we have to write two separate intervals.

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So, from negative infinity to zero, and five to infinity, which are two different

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sets.

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And how do we join them into one set?

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Did I teach you guys this?

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It's a U.

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Yeah.

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So we put the this big U looking thing

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stands for union

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all that means is

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These are two separate sets. To combine them into one set, put a union sign between

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them.

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Now this is one big set even though it's two separate intervals

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So is this an or or and?

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How about an and?

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Definitely, we see the word or.

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So the solutions are all the solutions to this, or all the solutions to this. For

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each inequality, we have to do a little work.

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of these we got to do a tiny bit of work.

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So we do divide by three.

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Subtract six.

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Fourteen.

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So solutions to the compound inequalities is everything we see in red.

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So in interval notation: negative infinity to 4, union 14 to infinity.

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Is this an and or an or?

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And.

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And how do we know?

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You know we got this inequality

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We have this inequality a

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Solution it's a solution to the compound inequality if it's a solution to the left

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inequality, and it's a solution to the right

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Okay, the beauty about these is we don't have to write two separate ones. We could

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solve it all at the same time

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So first is we subtract 6 from all three sides, which is really from each side of

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the two

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separate inequalities.

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So we get negative 10 is less than or equal to 2x, which is less than 4.

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Then we use the multiplication property of inequalities—or division—and divide all

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three sides by 2.

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by two, divide by two, divide by two.

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So this one, x is less than 2.

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That.

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This one, which same thing as x is greater than or equal to negative five.

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So the solution set has to be both in green and red here.

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So, that set of numbers, which is only one interval, so that's all the numbers

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between

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negative five and two excluding two, including negative five.

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Questions on this so far?

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So is the solution set where they overlap?

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Yes, but also when they overlap.

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Okay, it's not just where they overlap

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it's all the solutions to this which is everything in blue or

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Everything in purple whatever color that is

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Which is

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All real numbers

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We're gonna be careful just because it overlaps the solution set is not where it

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overlaps. It's very important

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And it's a solution to this, or it's a solution to this.

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So the solution set is all real numbers.

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This would be the graph of the solution set.

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In interval notation: negative infinity to positive infinity.

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So be real careful whether it's an or or an and.

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Just because it overlaps.

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That does not mean the solution set is only where they overlap.

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Okay.

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So is the solution set those two intervals?

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No.

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No.

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It has to be both in purple and in blue and there are no numbers in purple and blue.

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So what is the solution set?

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There are no solutions but what is the solution set?

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The empty set.

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Either that or we write it with curly brackets.

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Okay, be real careful.

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The AND and the OR make a big difference.

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There are no numbers, there are both solutions to this and solutions to this.

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So there are no solutions; the solution set is empty.

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Let me pass out the homework and stop here.