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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.