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