寫出與使用不等式 2 (英)

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寫出與使用不等式 2 (英) : 寫出與使用不等式 2

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J.T. loves burgers and loves to
subscribes to a cell phone texting plan with three other
members of his family.
Within any given month, they cannot send more than 500 text
messages total.
So they cannot send more than 500 text messages total.
At the end of this month, J.T.
had sent 25 more texts than his older sister.
Let me highlight.
Let me do this in different colors.
So at the end of the month, J.T.
had sent 25 more texts than his older sister, 50 fewer
texts than his younger sister, and 125 more
texts than his mother.
How may texts could J.T.
have sent if they did not go over the 500-text limit?
So let's just define some variables over here.
Well, let's say o.s.
o dot s is, well let's just say o, o is for older sister.
So this is number of texts by older sister.
Let's say that y is equal to number of texts
by his younger sister.
And then we'll use m is equal to the number of
texts by his mother.
And we'll use j for a number of texts by J.T.
So the total number of texts that everyone sent cannot be
more than 500.
So if we take the sum of J.T.'s texts, plus his older
sister's texts, plus his younger sister's texts, plus
his mother's texts, they all have to be less than or equal
to 500 total texts.
Right, it can't be more than 500, so the sum has to be less
than or equal to 500.
Now, how can we express each of these in terms of the
number of texts J.T.
sent?
Well, they give us some information here.
This first statement, J.T.
had sent 24 more texts than his older sister.
So j is equal to 25 plus the number of texts of his older
sister, which we say is o.
It's not a 0, that's an o for older sister.
And they also tell us that J.T.
sent 50 fewer texts than his younger sister, so j is also
equal to the younger sister minus 50, right?
50 fewer texts than his younger sister.
And then finally, they say 125 more texts than his mother, so
j is equal to mother plus 125.
Now, I want this equation all in terms of j's, because we
want to say how many texts could J.T.
have sent.
So I want all of these expressed in j, so let's just
solve each of these for o in terms of j, solve for y in
terms of j, solve for m in terms of j, and then we can
substitute back over here.
So if j is equal to 25 plus o, if we subtract 25 from both
sides of this equation, we get j minus 25 is equal to o.
These are the same thing.
If you just take this and subtract 25 from both sides,
you get that right there.
Now here, if you add 50 to both sides of this equation,
if you add to 50 to both sides of this equation, j plus 50 is
equal to the number of texts that his younger sister sent.
I just added 50 to both sides.
And then over here, if you subtract 125 from both sides
of this equation-- scroll over a little bit-- if you subtract
125 from both sides, you get j minus 125 is equal to the
number of texts sent by his mother.
And we could have gone straight here.
This first statement, J.T. has sent 25 more texts than is
older sister.
So if you take the number of texts J.T.
sent, subtracted 25, you'd get the number of texts by his
older sister.
That is an o, it is not a 0.
o for older.
Likewise, he sent 50 fewer texts than his younger sister,
so if you took the number of texts he sent, add 50 to it,
you're going to get how many his younger sister had sent.
And then finally, he sent 125 more than his mother, so if
you took J.T.'s texts, you take out 125, then that's how
many his mother sent.
So now that we have this, we can substitute for each of
these variables into the original equation.
So you have J.T.'s texts, plus his older sister's texts.
But we know that o is the same thing as J.T.'s texts minus
25, so we write J.T.'s texts minus 25.
And then you have plus his younger sister's texts, but we
know that's J.T.'s texts plus 50.
And then you have his mother's texts, but his mother's texts
are just J.T.'s texts minus 125, so plus j minus 125.
And all of that has to be less than or equal to 500.
So let's add the j's.
We have 1, 2, 3, 4 j's, so you have 4 j's.
And let's add the constants.
You have a negative 25 plus a 50, which is 25.
And then you have 25 minus 125, so 25 minus 125 is
negative 100.
So 4j minus 100-- I just added all the constant terms-- has
to be less than or equal to 500.
And now this is a pretty straightforward inequality.
Add 100 to both sides and we get 4j-- these cancel out-- we
get 4j is less than or equal to 600.
Divide both sides by 4-- don't have to worry about the
inequality since 4 is a positive number-- and we get j
is less than or equal to 150.
So J.T.
had to send less than or equal, he had to send 150 or
fewer texts in that month in order for all the constraints
to match up and for the family as a combined unit to send
less than 500.

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