Hi and welcome. My name is Anthony Varela. And this lesson is about compound inequalities. So we're going to talk about two types of inequalities-- one called and compound inequalities. Another called or compound inequalities. And then we'll talk about how to solve these two types of compound inequalities.
So first, let's talk about what a compound inequality is. Now, thinking back to elementary school, when you were learning about compound sentences, you learned these are two independent clauses that we could combine together. So for example, my favorite subject is math, and I like to swim. This is a compound sentence.
So with compound inequalities, we have two or more inequalities that we put together. So one example is negative 3 is less than x, and 5 is greater than x. So we see two inequality symbols. And we call this one in and compound inequality because we're connecting these two statements with the word and.
And here's another example-- negative 2 is less than x, or x is greater than 4. This is an example of an or compound inequality. We see more than one inequality symbol connected with the word or.
And you might also see an inequality that looks like this-- negative 3 is less than x, which is less than 5. So we have x in between two values. This is another way to write then an and compound inequality. So we're going to be talking about two types of compound inequalities, and and or.
So let's talk more about and compound inequalities. So here's an example. x is greater than negative 4, and x is less than 3. So thinking about solutions to this compound inequality, well, here is one-- x is 2.
So if I put 2 in for x, what do I get? Well, I get the statement that 2 is greater than negative 4, and 2 is less than 3. And this is true. 2 is both greater than negative 4 and less than 3. So our solutions to and compound inequalities must satisfy both, or all, I should say, inequalities that make up the compound inequality.
So putting this on a number line, we can write down all x values that are greater than negative 4 and all x values that are less than 3. And what we're looking for is the range of values that satisfies both of these two inequalities here. So to simplify our number line, this would represent the solution to my and compound inequality.
And I can write this as negative 4 is less than x, which is less than 3. And this also makes up the inequality that goes into set notation. And my interval, then, is from negative 4 to 3. I almost forgot that. So I'll fix that right there. So with our and compound inequalities, our solutions must satisfy all of the inequalities that make up the compound inequality.
So now, let's talk about or compound inequalities. Here's an example of that. x is less than 2, or x is greater than 7. So thinking about a solution to this or compound inequality, I could say x equals negative 3. So if I put in negative 3 then for x, what do I get?
I get negative 3 is less than 2, which is true. And then I have negative 3 is greater than 7, which is not true. So we see that this satisfies one or the other, but not necessarily both. And if we had more than two, this would be satisfying at least one of our individual inequalities, not necessarily all of them.
So x equals 8 would be another example of a solution. It satisfies one of my inequalities but not the other. And that's OK with an or compound inequality. So let's put this then on the number line.
So here is all x values that are less than 2. And then we have all x values that are greater than 7. So we see that there is no x value that could satisfy both. That's just impossible. We see that on our number line. You either have x values that are less than 2 or greater than 7.
So we've created, then, two intervals. So writing this solution, then, in set notation would be all x such that x is less than 2, or x is greater than 7. And then we can have two intervals-- negative infinity to 2 and then 7 to positive infinity. And we connect them with our union symbol with or compound inequalities. So they satisfy at least one inequality.
All right, now, let's talk about solving compound inequalities. So here is our example. We have negative 2 is less than 3x plus 7, which is less than 13. So we have this quantity 3x plus 7 that's in between negative 2 and 13. Now, taking a look at 3x plus 7, and if this are part of a linear equation, I am familiar with solving this. And with our compound inequalities, just remember whatever you do to one portion or one section of this compound inequality, you do to everything else.
So here's how I'm going to start by solving. I'm going to take away 7 so that I have just 3x in my middle section. But then I have to take away 7 in all my other parts as well. So rewriting this, I now have negative 9 is less than 3x, which is less than 6, subtracting 7 across the entire compound inequality.
Next, I'm going to divide by 3 to isolate x, but I'm going to do that to my other sections here. So now, I have negative 3 is less than x, which is less than 2 when I divided all sections by 3. So now I have something that I can plot on the number line and then write its solutions in a couple of different notations.
So let's pull our number line. I'm going to mark down negative 3 and 2. And I'm not including those exact values, so I have my open circles. And in set notation, this is all x, such that x is in between negative 3 and positive 2. And in interval notation, this goes from negative 3 to positive 2, not including those two values.
All right. Now, let's solve an or compound inequality. So here, I have 2x minus 5 is less than 7, or 2x minus 5 is greater than 3. So here, I'm going to be solving these two inequalities separately.
So first, I'm going to add 5 to both sides here. And I'm going to add 5 to both sides here, because we just so happen to have 2x minus 5 as part of both of these inequalities. So now, what I have is 2x is less than negative 2, or 2x is greater than 8.
Well, I'm going to divide both sides by 2 here and divide both sides by 2 here. And now, I have x is less than negative 1, or x is greater than 4. So let's go ahead then and plot this on the number line. So I'm bringing up my number line. I'm going to write down all x values that are less than negative 1, not including negative 1.
And I'm also going to highlight then all x values that are greater than 4, but not including 4. So here's that written in set notation. All x such that x is less than negative 1, or x is greater than 4. And in interval notation, we have negative infinity to negative 1. And we also have 4 to positive infinity. And I'm going to put in my union symbol.
So let's review solving compound inequalities. Well, we talked about and and or compound inequalities. With our and compound inequalities, the solutions must satisfy all inequalities that make up the compound inequality. But with our or compound inequalities, the solutions must satisfy at least one, not necessarily all of them.
Here's how we see them connected with and and or. You might also see and compound inequalities with x in between two values. And this is what their solutions look like on the number line. Here, we have a range between A and B. And here, we have two intervals, one or the other. That's part of our solution.
So thanks for watching this tutorial on compound inequalities. Hope to see you next time.