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We have the inequality 2/3 is greater than negative

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4y minus 8 and 1/3.

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Now, the first thing I want to do here, just because mixed

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numbers bother me-- they're actually hard to deal with

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mathematically.

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They're easy to think about-- oh, it's a little

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bit more than 8.

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Let's convert this to an improper fraction.

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So 8 and 1/3 is equal to-- the denominator's going to be 3.

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3 times 8 is 24, plus 1 is 25.

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So this thing over here is the same thing as 25 over 3.

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Let me just rewrite the whole thing.

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So it's 2/3 is greater than negative 4y minus 25 over 3.

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Now, the next thing I want to do, just because dealing with

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fractions are a bit of a pain, is multiply both sides of this

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inequality by some quantity that'll

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eliminate the fractions.

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And the easiest one I can think of is multiply both

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sides by 3.

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That'll get rid of the 3's in the denominator.

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So let's multiply both sides of this equation by 3.

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That's the left-hand side.

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And then I'm going to multiply the right-hand side.

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3, I'll put it in parentheses like that.

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Well, one point that I want to point out is that I did not

27
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have to swap the inequality sign, because I multiplied

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both sides by a positive number.

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If the 3 was a negative number, if I multiplied both

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sides by negative 3, or negative 1, or negative

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whatever, I would have had to swap the inequality sign.

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Anyway, let's simplify this.

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So the left-hand side, we have 3 times 2/3, which is just 2.

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2 is greater than.

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And then we can distribute this 3.

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3 times negative 4y is negative 12y.

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And then 3 times negative 25 over 3 is just negative 25.

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Now, we want to get all of our constant terms on one side of

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the inequality and all of our variable terms-- the only

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variable here is y on the other side-- the y is already

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sitting here, so let's just get this 25 on the other side

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of the inequality.

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And we can do that by adding 25 to both

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sides of this equation.

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So let's add 25 to both sides of this equation.

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And with the left-hand side, 2 plus 25 five is 27 and we're

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going to get 27 is greater than.

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The right-hand side of the inequality is negative 12y.

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And then negative 25 plus 25, those cancel out, that was the

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whole point, so we're left with 27 is greater than

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negative 12y.

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Now, to isolate the y, you can either multiply both sides by

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negative 1/12 or you could say let's just divide both sides

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by negative 12.

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Now, because I'm multiplying or dividing by a negative

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number here, I'm going to need to swap the inequality.

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So let me write this.

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If I divide both sides of this equation by negative 12, then

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it becomes 27 over negative 12 is less than-- I'm swapping

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the inequality, let me do this in a different color-- is less

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than negative 12y over negative 12.

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Notice, when I divide both sides of the inequality by a

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negative number, I swap the inequality, the greater than

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becomes a less than.

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When it was positive, I didn't have to swap it.

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So 27 divided by negative 12, well, they're both

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divisible by 3.

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So we're going to get, if we divide the numerator and the

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denominator by 3, we get negative 9 over 4 is less

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than-- these cancel out-- y.

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So y is greater than negative 9/4, or negative 9/4

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is less than y.

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And if you wanted to write that-- just let me write

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this-- our answer is y is greater than negative 9/4.

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I just swapped the order, you could say negative 9/4

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is less than y.

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Or if you want to visualize that a little bit better, 9/4

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is 2 and 1/4, so we could also say y is greater than negative

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2 and 1/4 if we want to put it as a mixed number.

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And if we wanted to graph it on the number line-- let me

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draw a number line right here, a real simple one.

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Maybe this is 0.

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Negative 2 is right over, let's say negative 1, negative

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2, then say negative 3 is right there.

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Negative 2 and 1/4 is going to be right here, and it's

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greater than, so we're not going to include that in the

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solution set.

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So we're going to make an open circle right there.

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And everything larger than that is a valid y, is a y that

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will satisfy the inequality.

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