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- In this video, I want to tackle some inequalities that
- involve multiplying and dividing by positive and
- negative numbers, and you'll see that it's a little bit
- more tricky than just the adding and subtracting numbers
- that we saw in the last video.
- I also want to introduce you to some other types of
- notations for describing the solution set of an inequality.
- So let's do a couple of examples.
- So let's say I had negative 0.5x is less
- than or equal to 7.5.
- Now, if this was an equality, your natural impulse is to
- say, hey, let's divide both sides by the coefficient on
- the x term, and that is a completely legitimate thing to
- do: divide both sides by negative 0.5.
- The important thing you need to realize, though, when you
- do it with an inequality is that when you multiply or
- divide both sides of the equation by a negative number,
- you swap the inequality.
- you swap the inequality
- Think of it this way.
- I'll do a simple example here.
- If I were to tell you that 1 is less than 2, I think you
- would agree with that.
- 1 is definitely less than 2.
- Now, what happens if I multiply both sides of this by
- negative 1?
- Negative 1 versus negative 2?
- Well, all of a sudden, negative 2 is more negative
- than negative 1.
- So here, negative 2 is actually less than negative 1.
- Now, this isn't a proof, but I think it'll give you comfort
- on why you're swapping the sign.
- If something is larger, when you take the negative of both
- of it, it'll be more negative, or vice versa.
- So that's why, if we're going to multiply both sides of this
- equation or divide both sides of the equation by a negative
- number, we need to swap the sign.
- So let's multiply both sides of this equation.
- Dividing by 0.5 is the same thing as multiplying by 2.
- Our whole goal here is to have a 1 coefficient there.
- So let's multiply both sides of this
- equation by negative 2.
- So we have negative 2 times negative 0.5.
- And you might say, hey, how did Sal get this 2 here?
- My brain is just thinking what can I multiply negative
- 0.5 by to get 1?
- And negative 0.5 is the same thing as negative 1/2.
- The inverse of that is negative 2.
- So I'm multiplying negative 2 times both
- sides of this equation.
- And I have the 7.5 on the other side.
- I'm going to multiply that by negative 2 as well.
- And remember, when you multiply or divide both sides
- of an inequality by a negative, you swap the
- inequality.
- You had less than or equal?
- Now it'll be greater than or equal.
- So the left-hand side, negative 2 times
- negative 0.5 is just 1.
- You get x is greater than or equal to 7.5 times negative 2.
- That's negative 15, which is our solution set.
- All x's larger than negative 15 will satisfy this equation.
- I challenge you to try it.
- For example, 0 will work.
- 0 is greater than negative 15.
- But try something like-- try negative 16.
- Negative 16 will not work.
- Negative 16 times negative 0.5 is 8, which is
- not less than 7.5.
- So the solution set is all of the x's-- let me draw a number
- line here-- greater than negative 15.
- So that is negative 15 there, maybe that's negative 16,
- that's negative 14.
- Greater than or equal to negative 15 is the solution.
- Now, you might also see solution sets to inequalities
- written in interval notation.
- And interval notation, it just takes a little
- getting used to.
- We want to include negative 15, so our lower bound to our
- interval is negative 15.
- And putting in this bracket here means that we're going to
- include negative 15.
- The set includes the bottom boundary.
- It includes negative 15.
- And we're going to go all the way to infinity.
- All the way to infinity
- And we put a parentheses here.
- Parentheses normally means that you're not including the
- upper bound.
- You also do it for infinity, because infinity really isn't
- a normal number, so to speak.
- You can't just say, oh, I'm at infinity.
- You're never at infinity.
- So that's why you put that parentheses.
- But the parentheses tends to mean that you don't include
- that boundary, but you also use it with infinity.
- So this and this are the exact same thing.
- Sometimes you might also see set notations, where the
- solution of that, they might say x is a real number such
- that-- that little line, that vertical line thing, just
- means such that-- x is greater than or equal to negative 15.
- These curly brackets mean the set of all real numbers, or
- the set of all numbers, where x is a real number, such that
- x is greater than or equal to negative 15.
- All of this, this, and this are all equivalent.
- Let's keep that in mind and do a couple of more examples.
- So let's say we had 75x is greater than or equal to 125.
- So here we can just divide both sides by 75.
- And since 75 is a positive number, you don't have to
- change the inequality.
- So you get x is greater than or equal to 125/75.
- And if you divide the numerator and denominator by
- 25, this is 5/3.
- So x is greater than or equal to 5/3.
- Or we could write the solution set being from including 5/3
- to infinity.
- And once again, if you were to graph it on a number
- line, 5/3 is what?
- That's 1 and 2/3.
- So you have 0, 1, 2, and 1 and 2/3 will be
- right around there.
- We're going to include it.
- That right there is 5/3.
- And everything greater than or equal to that will be included
- in our solution set.
- Let's do another one.
- Let's say we have x over negative 3 is greater than
- negative 10/9.
- So we want to just isolate the x on the left-hand side.
- So let's multiply both sides by negative 3, right?
- The coefficient, you could imagine, is negative 1/3, so
- we want to multiply by the inverse, which should be
- negative 3.
- So if you multiply both sides by negative 3, you get
- negative 3 times-- this you could rewrite it as negative
- 1/3x, and on this side, you have negative 10/9 times
- negative 3.
- And the inequality will switch, because we are
- multiplying or dividing by a negative number.
- So the inequality will switch.
- It'll go from greater than to less than.
- So the left-hand side of the equation just becomes an x.
- That was the whole point.
- That cancels out with that.
- The negatives cancel out. x is less than.
- And then you have a negative times a negative.
- That will make it a positive.
- Then if you divide the numerator and the denominator
- by 3, you get a 1 and a 3, so x is less than 10/3.
- So if we were to write this in interval notation, the
- solution set will-- the upper bound will be 10/3 and it
- won't include 10/3.
- This isn't less than or equal to, so we're going to put a
- parentheses here.
- Notice, here it included 5/3.
- We put a bracket.
- Here, we're not including 10/3.
- We put a parentheses.
- It'll go from 10/3, all the way down to negative infinity.
- Everything less than 10/3 is in our solution set.
- And let's draw that.
- Let's draw the solution set.
- So 10/3, so we might have 0, 1, 2, 3, 4.
- 10/3 is 3 and 1/3, so it might sit-- let me do it in a
- different color.
- It might be over here.
- We're not going to include that.
- It's less than 10/3.
- 10/3 is not in the solution set.
- That is 10/3 right there, and everything less than that, but
- not including 10/3, is in our solution set.
- Let's do one more.
- Let's do one more.
- Say we have x over negative 15 is less than 8.
- So once again, let's multiply both sides of this equation by
- negative 15.
- So negative 15 times x over negative 15.
- Then you have an 8 times a negative 15.
- And when you multiply both sides of an inequality by a
- negative number or divide both sides by a negative number,
- you swap the inequality.
- It's less than, you change it to greater than.
- And now, this left-hand side just becomes an x, because
- these guys cancel out.
- x is greater than 8 times 15 is 80 plus 40 is 120, so
- negative 120.
- Is that right?
- 80 plus 40.
- Yep, negative 120.
- Or we could write the solution set as starting at negative
- 120-- but we're not including negative 120.
- We don't have an equal sign here-- going all
- the way up to infinity.
- And if we were to graph it, let me draw
- the number line here.
- I'll do a real quick one.
- Let's say that that is negative 120.
- Maybe zero is sitting up here.
- This would be negative 121.
- This would be negative 119.
- We are not going to include negative 120, because we don't
- have an equal sign there, but it's going to be everything
- greater than negative 120.
- All of these things that I'm shading in green would satisfy
- the inequality.
- And you can even try it out.
- Does zero work?
- 0/15?
- Yeah, that's zero.
- That's definitely less than 8.
- I mean, that doesn't prove it to you, but you could try any
- of these numbers and they should work.
- Anyway, hopefully, you found that helpful.
- I'll see you in the next video.