解不等式 (英)

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解不等式 (英) : 解不等式

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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
mathematically.
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.

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