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In this video, I want to tackle some inequalities that

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involve multiplying and dividing by positive and

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negative numbers, and you'll see that it's a little bit

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more tricky than just the adding and subtracting numbers

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that we saw in the last video.

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I also want to introduce you to some other types of

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notations for describing the solution set of an inequality.

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So let's do a couple of examples.

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So let's say I had negative 0.5x is less

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than or equal to 7.5.

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Now, if this was an equality, your natural impulse is to

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say, hey, let's divide both sides by the coefficient on

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the x term, and that is a completely legitimate thing to

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do: divide both sides by negative 0.5.

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The important thing you need to realize, though, when you

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do it with an inequality is that when you multiply or

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divide both sides of the equation by a negative number,

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you swap the inequality.

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Think of it this way.

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I'll do a simple example here.

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If I were to tell you that 1 is less than 2, I think you

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would agree with that.

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1 is definitely less than 2.

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Now, what happens if I multiply both sides of this by

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negative 1?

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Negative 1 versus negative 2?

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Well, all of a sudden, negative 2 is more negative

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than negative 1.

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So here, negative 2 is actually less than negative 1.

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Now, this isn't a proof, but I think it'll give you comfort

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on why you're swapping the sign.

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If something is larger, when you take the negative of both

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of it, it'll be more negative, or vice versa.

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So that's why, if we're going to multiply both sides of this

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equation or divide both sides of the equation by a negative

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number, we need to swap the sign.

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So let's multiply both sides of this equation.

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Dividing by 0.5 is the same thing as multiplying by 2.

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Our whole goal here is to have a 1 coefficient there.

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So let's multiply both sides of this

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equation by negative 2.

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So we have negative 2 times negative 0.5.

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And you might say, hey, how did Sal get this 2 here?

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My brain is just thinking what can I multiply negative

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0.5 by to get 1?

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And negative 0.5 is the same thing as negative 1/2.

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The inverse of that is negative 2.

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So I'm multiplying negative 2 times both

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sides of this equation.

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And I have the 7.5 on the other side.

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I'm going to multiply that by negative 2 as well.

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And remember, when you multiply or divide both sides

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of an inequality by a negative, you swap the

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inequality.

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You had less than or equal?

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Now it'll be greater than or equal.

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So the left-hand side, negative 2 times

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negative 0.5 is just 1.

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You get x is greater than or equal to 7.5 times negative 2.

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That's negative 15, which is our solution set.

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All x's larger than negative 15 will satisfy this equation.

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I challenge you to try it.

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For example, 0 will work.

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0 is greater than negative 15.

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But try something like-- try negative 16.

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Negative 16 will not work.

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Negative 16 times negative 0.5 is 8, which is

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not less than 7.5.

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So the solution set is all of the x's-- let me draw a number

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line here-- greater than negative 15.

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So that is negative 15 there, maybe that's negative 16,

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that's negative 14.

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Greater than or equal to negative 15 is the solution.

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Now, you might also see solution sets to inequalities

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written in interval notation.

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And interval notation, it just takes a little

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getting used to.

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We want to include negative 15, so our lower bound to our

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interval is negative 15.

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And putting in this bracket here means that we're going to

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include negative 15.

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The set includes the bottom boundary.

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It includes negative 15.

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And we're going to go all the way to infinity.

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And we put a parentheses here.

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Parentheses normally means that you're not including the

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upper bound.

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You also do it for infinity, because infinity really isn't

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a normal number, so to speak.

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You can't just say, oh, I'm at infinity.

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You're never at infinity.

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So that's why you put that parentheses.

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But the parentheses tends to mean that you don't include

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that boundary, but you also use it with infinity.

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So this and this are the exact same thing.

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Sometimes you might also see set notations, where the

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solution of that, they might say x is a real number such

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that-- that little line, that vertical line thing, just

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means such that-- x is greater than or equal to negative 15.

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These curly brackets mean the set of all real numbers, or

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the set of all numbers, where x is a real number, such that

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x is greater than or equal to negative 15.

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All of this, this, and this are all equivalent.

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Let's keep that in mind and do a couple of more examples.

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So let's say we had 75x is greater than or equal to 125.

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So here we can just divide both sides by 75.

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And since 75 is a positive number, you don't have to

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change the inequality.

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So you get x is greater than or equal to 125/75.

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And if you divide the numerator and denominator by

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25, this is 5/3.

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So x is greater than or equal to 5/3.

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Or we could write the solution set being from including 5/3

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to infinity.

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And once again, if you were to graph it on a number

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line, 5/3 is what?

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That's 1 and 2/3.

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So you have 0, 1, 2, and 1 and 2/3 will be

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right around there.

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We're going to include it.

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That right there is 5/3.

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And everything greater than or equal to that will be included

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in our solution set.

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Let's do another one.

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Let's say we have x over negative 3 is greater than

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negative 10/9.

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So we want to just isolate the x on the left-hand side.

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So let's multiply both sides by negative 3, right?

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The coefficient, you could imagine, is negative 1/3, so

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we want to multiply by the inverse, which should be

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negative 3.

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So if you multiply both sides by negative 3, you get

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negative 3 times-- this you could rewrite it as negative

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1/3x, and on this side, you have negative 10/9 times

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negative 3.

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And the inequality will switch, because we are

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multiplying or dividing by a negative number.

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So the inequality will switch.

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It'll go from greater than to less than.

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So the left-hand side of the equation just becomes an x.

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That was the whole point.

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That cancels out with that.

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The negatives cancel out. x is less than.

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And then you have a negative times a negative.

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That will make it a positive.

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Then if you divide the numerator and the denominator

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by 3, you get a 1 and a 3, so x is less than 10/3.

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So if we were to write this in interval notation, the

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solution set will-- the upper bound will be 10/3 and it

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won't include 10/3.

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This isn't less than or equal to, so we're going to put a

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parentheses here.

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Notice, here it included 5/3.

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We put a bracket.

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Here, we're not including 10/3.

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We put a parentheses.

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It'll go from 10/3, all the way down to negative infinity.

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Everything less than 10/3 is in our solution set.

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And let's draw that.

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Let's draw the solution set.

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So 10/3, so we might have 0, 1, 2, 3, 4.

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10/3 is 3 and 1/3, so it might sit-- let me do it in a

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different color.

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It might be over here.

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We're not going to include that.

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It's less than 10/3.

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10/3 is not in the solution set.

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That is 10/3 right there, and everything less than that, but

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not including 10/3, is in our solution set.

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Let's do one more.

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Say we have x over negative 15 is less than 8.

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So once again, let's multiply both sides of this equation by

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negative 15.

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So negative 15 times x over negative 15.

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Then you have an 8 times a negative 15.

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And when you multiply both sides of an inequality by a

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negative number or divide both sides by a negative number,

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you swap the inequality.

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It's less than, you change it to greater than.

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And now, this left-hand side just becomes an x, because

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these guys cancel out.

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x is greater than 8 times 15 is 80 plus 40 is 120, so

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negative 120.

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Is that right?

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80 plus 40.

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Yep, negative 120.

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Or we could write the solution set as starting at negative

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120-- but we're not including negative 120.

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We don't have an equal sign here-- going all

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the way up to infinity.

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And if we were to graph it, let me draw

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the number line here.

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I'll do a real quick one.

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Let's say that that is negative 120.

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Maybe zero is sitting up here.

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This would be negative 121.

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This would be negative 119.

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We are not going to include negative 120, because we don't

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have an equal sign there, but it's going to be everything

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greater than negative 120.

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All of these things that I'm shading in green would satisfy

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the inequality.

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And you can even try it out.

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Does zero work?

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0/15?

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Yeah, that's zero.

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That's definitely less than 8.

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I mean, that doesn't prove it to you, but you could try any

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of these numbers and they should work.

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Anyway, hopefully, you found that helpful.

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I'll see you in the next video.

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