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解不等式 (英) : 解不等式
- We have the inequality 2/3 is greater than negative
- 4y minus 8 and 1/3.
- Now, the first thing I want to do here, just because mixed
- numbers bother me-- they're actually hard to deal with
- They're easy to think about-- oh, it's a little
- bit more than 8.
- Let's convert this to an improper fraction.
- So 8 and 1/3 is equal to-- the denominator's going to be 3.
- 3 times 8 is 24, plus 1 is 25.
- So this thing over here is the same thing as 25 over 3.
- Let me just rewrite the whole thing.
- So it's 2/3 is greater than negative 4y minus 25 over 3.
- Now, the next thing I want to do, just because dealing with
- fractions are a bit of a pain, is multiply both sides of this
- inequality by some quantity that'll
- eliminate the fractions.
- And the easiest one I can think of is multiply both
- sides by 3.
- That'll get rid of the 3's in the denominator.
- So let's multiply both sides of this equation by 3.
- That's the left-hand side.
- And then I'm going to multiply the right-hand side.
- 3, I'll put it in parentheses like that.
- Well, one point that I want to point out is that I did not
- have to swap the inequality sign, because I multiplied
- both sides by a positive number.
- If the 3 was a negative number, if I multiplied both
- sides by negative 3, or negative 1, or negative
- whatever, I would have had to swap the inequality sign.
- Anyway, let's simplify this.
- So the left-hand side, we have 3 times 2/3, which is just 2.
- 2 is greater than.
- And then we can distribute this 3.
- 3 times negative 4y is negative 12y.
- And then 3 times negative 25 over 3 is just negative 25.
- Now, we want to get all of our constant terms on one side of
- the inequality and all of our variable terms-- the only
- variable here is y on the other side-- the y is already
- sitting here, so let's just get this 25 on the other side
- of the inequality.
- And we can do that by adding 25 to both
- sides of this equation.
- So let's add 25 to both sides of this equation.
- --Adding 25--
- And with the left-hand side, 2 plus 25 five is 27 and we're
- going to get 27 is greater than.
- The right-hand side of the inequality is negative 12y.
- And then negative 25 plus 25, those cancel out, that was the
- whole point, so we're left with 27 is greater than
- negative 12y.
- Now, to isolate the y, you can either multiply both sides by
- negative 1/12 or you could say let's just divide both sides
- by negative 12.
- Now, because I'm multiplying or dividing by a negative
- number here, I'm going to need to swap the inequality.
- So let me write this.
- If I divide both sides of this equation by negative 12, then
- it becomes 27 over negative 12 is less than-- I'm swapping
- the inequality, let me do this in a different color-- is less
- than negative 12y over negative 12.
- Notice, when I divide both sides of the inequality by a
- negative number, I swap the inequality, the greater than
- becomes a less than.
- When it was positive, I didn't have to swap it.
- So 27 divided by negative 12, well, they're both
- divisible by 3.
- So we're going to get, if we divide the numerator and the
- denominator by 3, we get negative 9 over 4 is less
- than-- these cancel out-- y.
- So y is greater than negative 9/4, or negative 9/4
- is less than y.
- And if you wanted to write that-- just let me write
- this-- our answer is y is greater than negative 9/4.
- I just swapped the order, you could say negative 9/4
- is less than y.
- Or if you want to visualize that a little bit better, 9/4
- is 2 and 1/4, so we could also say y is greater than negative
- 2 and 1/4 if we want to put it as a mixed number.
- And if we wanted to graph it on the number line-- let me
- draw a number line right here, a real simple one.
- Maybe this is 0.
- Negative 2 is right over, let's say negative 1, negative
- 2, then say negative 3 is right there.
- Negative 2 and 1/4 is going to be right here, and it's
- greater than, so we're not going to include that in the
- solution set.
- So we're going to make an open circle right there.
- And everything larger than that is a valid y, is a y that
- will satisfy the inequality.