Solve Absolute Value Inequalities - Example 1
Solve Absolute Value Inequalities - Example 2
Solve Absolute Value Inequalities - Example 3
Solve Absolute Value Inequalities - Example 4
Solve Absolute Value Inequalities - Overview
Syracuse University
Graphing and Writing Inequalities - Overview


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So in this next chapter, we're gonna be talking about how to solve and graph inequalities. Now, in the next couple of sections, you'll end up finding out How do we solve inequalities? What you'll find is very similar to solving equations. But if you remember back from Chapter One, we talked about how there's multiple solutions to inequalities. So when we go ahead and represent their solution, what we're going to do is graft. Their solution sets on a number line. Now, just as a couple of quick reminders, we have our four inequality symptoms, as I have listed here in our little table. So remember, we have our lesson symbol less than or equal to greater than and greater than or equal to. So what we're gonna talk about in this section as how we can actually graph these solution sets on a number line. So let's go to our examples. So our first example is that X is less than two. So the question is, how would we represent all the values that fit the solution set? So in this case, we're saying X is less than two meaning, meaning any number less than two with it the solution set. So if you think about numbers that are less than two, you would think you're thinking about 10 Negative three negative five negative 5000. All of these different values for acts would make this inequality true. So now the question is, Well, how do we represent us on the number line? So the first thing is, but I typically call our key point is our starting point, which always happens at the value for our solution at two. So the question is, is to actually a solution to this inequality. Well, let's look, if we substituted to infer acts, we would have to is less than two. Well, is that true or false? Well, this would be false because two isn't less than two. It's equal to it. But it's not less than so. This is going to be false. However, with 1.9 b a solution. Well, yes, it would, because 1.9 is less than two. So, essentially, we're gonna have all the values up till to just not including two. So to represent that too is not a solution. What we're gonna do on our number line at two. Is we're gonna want to put an open circle there. So an open circle means that it's up till that point, just not including it. Now, what we're gonna do is we're gonna shade all the possible values that X could possibly be. Well, like we said, it's any value of X. That's less than two. So if you think about your number, line where the values less than two well, there anywhere, any place to the left of two. So what we're gonna do on our number line is we're going to shade. So we're in the shade all the way down to the arrow, and then I typically kind of shade over the arrow really to emphasize that we're talking about all these values that are less than two. So what this means is any of these values in the shaded area will represent possible solutions to our inequality. X is less than two. All right, let's try the next one. It says that X is less than or equal to two. Well, many of these solutions are the same as the previous one, because it's going to include all the values that are less than to accept now in this case we have to think about is to included Well, let's think about it is to less than or equal to two. Well, this is a true statement because two is equal to two. So what Fists? The or equal to part. So whenever you see a lesson or equal to sign, really just or equal to sign at two were going toe use a close circle. So the close circle means that, too, is included in the solution set. Now we need to shave all the other possible values that could make this true, just like with the previous example, because we're looking for all the values of X that are less than or equal to to all these values would be to the left of two on our number line. So just like in the previous example, we're going to shade all these values of X that are less than two on our number line, meaning that they all follow to the left. And I think my markets being a little funny here. So again, we're shading all these values to the left. It too, so notice when you have X, is less than a value Oh, let's get rid of that undoing. There we go. So notice any time you have X is less than the value, you end up shading to the left, and that will always be true if you're variable comes first. Now let's talk about example. Number three we have X is greater than two. So again, the first thing we have to determine is what kind of circle happens to so first. That we have to ask ourselves is to a solution. Well is to greater than two. Well, that would be false to is not greater than two, which means that at two we're going to have an open circle. So again, the one thing hopefully you've noticed is when you don't have a more equal to sign. That's when you're going to use an open circle. So, like I said at two will have an open circle. So now we have to determine should be shade to the left or to the right of to Well, now we're talking about all the values that are greater than two, so values that are greater than two would be like 345 700. While all of these values on our number line would occur to the right of two. So what we're gonna do in this case is we're gonna shade on our number line to the right of to. So this shaded area represents all the possible values of X that will make it true. And now, for our gods, example X is greater than or equal to two again. The first thing you should ask yourself is, should it be an open or closed circle? Meaning is to a solution. Well is to greater than or equal to two. Well, this is true because two is equal to two, so therefore it's greater than or equal to it. So at two, we're going to use a close circle. And just like in the previous example, because we're talking about all the values that are greater than or equal to two, we're going to shade to the right on our number line. So notice when we're talking about values of X that are greater than the number we're going to shade to the right, as opposed to in the first two examples where we're talking about values of X that are less than X. We're gonna be shading to the left. The other thing to note is, in terms of one. Should we have an open circle or a close circle? What, as you can see if it's just a lesson sign or a greater than sign, this is when you're gonna have an open circle at that particular value because it means that it's everything up until that particular number, just not including it. However, when you have a less than or equal to sign or if you have a greater than or equal to sign, this is when you're going to use a close circle because this will represent that it could be that specific value as well. So in the next couple examples, we're gonna go over some, or how you can graph on the number line and go over some special cases as well as maybe you're given the number line and you need to write in inequality to represent that solution. Set

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