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Solve Compound Inequalities - Or Statements - Overview

In mathematics, a compound inequality is an inequality involving two or more inequalities connected by logical operators such as "and" and "or". Compound inequalities can be written in many ways. For example: 2x+3 or 5x+5 >10

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in this section, we're going to start talking about compound inequalities. And between the next two sections, you're gonna find that there's two different types and it's based on ah, given word. We're even gonna have what we're going to talk about in this section, which is called, or statements or the other type of inequality compound inequalities that we could have our and statements. Like I said for this section, we're just going to talk about or statements. So let's think about what is the order where the or mean Well, typically it's used in cases like either or so. For example, if I said John or Bob went to the store, that means even John Could went to the store or Bob went to the store. Not that they both had to go to the store. To make that sense is true. So let's take it the the phrase that we have here. It says all real numbers less than negative chew or greater than three. So we're gonna end up writing an inequality for this. But it might be helpful to think about what would this look like on a number line for the solution sets so let's start with the first part. It says all real numbers that are less than negative, too. Well, would that mean negative two could be included in a solution? It can. So that would be a negative to we would have an open circle. Well, if we're talking about all the real numbers that are less than negative to, that would be like negative three negative four and negative five. So we would end up shaving to the left hand side of our circle. So all these values in the shaded area would represent all of the real numbers that are less than negative, too. Just not including negative, too. That's why we have the open circle there. Okay, let's now think about the second part of our segments, it says, Or it could be greater than three, so this value could be bigger than three. It can't be three because it didn't say greater than or equal to. So in regards to our number line, we would have an open circle it three. Well, let's think about it where the values are. Where are real numbers that are greater than three? Well, that would occur to the right on the number line so we would shade the outside section. So let's think about this in terms of our sentence. It says all real numbers that are less than negative to or greater than three. So think about what we didn't shave this middle section. All these values in between are bigger than negative, too. But that's why they don't fit and they're not bigger than three. So that's why the section goes unshaded. Okay, well, now let's go ahead and try and write and inequality to represent the statement, and it might be helpful to use your number line. So let's just pretend this first section was the only thing on our number line well have would be right in inequality. Well, we we start with our variable, which is X. We shaded the left hand side, so we're going to be less than we're not gonna have your equal to, because it's open circle and our key value is negative. Two. So this left half section is represented by X is less than negative, too. Well, we have another section to our number line. We have this section here to the right, so let's go ahead and write it in the quality to represent it. So we will start with X, are key value is three because we shared it to the right is going to be greater than and it's not including three, so we will have the or equal to part. So now the Onley everything that we need is to represent that it's an or statement. So to do this, all we do is rewrite the word or between them. So what we've done is we've now written a compound inequality. So key thing is, when you're using the word or it means either one could be true. So as long as you have values for X that make either of these two inequality statements true, they will be a part of your solution set. That's essentially what a compound inequality means. Okay, so now the question is, Well, how do we go ahead and solve these compounds in the qualities? Well, notice In this example, we have three X is greater than 12 or X minus two is less than negative three. So the first to go about solving them you solve both inequality separate, almost as if they're their own problems. So how do we fight? Solve our first inequality? Well, we would simply divide both sides by three, so we would be left with X is greater than while 12, divided by three is four. Now we'll go to our second inequality. Rx minus two is less than three less than negative three. So the salt, this inequality, we're gonna add to to both sides because the twos will cancel. So we're left with X is less then we'll negative three plus two is negative one and then I can bring down my or statement. So what we found is our values of acts that will make this true are any values of acts that are greater than for or less than negative ones. So now how do we graft us on your number line? Well, we're going to start by setting up our number line now where you've been used to just putting one key number on our number line. But in this case, we have two key numbers. We have Ford and negative one. Make sure you put the smaller of them on the left hand side and the larger them on the right hand side, just like you would for a regular number line. So here's how I teach my students autographed ease. When it's an or statement, it means that either one could be true. So essentially, the best way to go ahead and draftees is to graft them both separately. Meaning, How do we graph X is greater than four? Well, we would have an open circle up for and we would shade to the right, So that means anything in this blue section will make this first inequality true. Then we're going to graph our second inequality. X is less than negative one well, to graft this, we would have an open circle it negative one, and we would shade everything to the left, meaning any value in this green shaded area will make our second inequality true. And because it's an or statement as long as it makes either in the quality true, the whole thing will be true. So my recommendation when it comes to actually graphing compound inequalities that involved or on a number line, all you need to do is graft them both separately, just on the same number line. Now you'll find in the next couple of examples that we dio that there are some special cases when it comes to the number line. Um, so it's really important that you just kind of treat both of those inequalities separately and you won't have to worry about a special cases. So, like I said, stay tuned to the next couple of examples and we'll go over how to graft Mawr compound inequalities.

Syracuse University
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