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November 11th, 2018, 01:20 PM   #1
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simple inequality problem

x and y integer
-5<x<4
-8<y<1

what is the maximum value of 2x-3y?

when i multiply first eqn by 2 and second by -3 and the add them up i get
-13<2x-3y<32 which makes the answer 31 but that is not even given in answer choices and the answer they say should be 27 . please help
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November 11th, 2018, 01:47 PM   #2
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Quote:
Originally Posted by ketanco View Post
x and y integer
-5<x<4
-8<y<1

what is the maximum value of 2x-3y?

when i multiply first eqn by 2 and second by -3 and the add them up i get
-13<2x-3y<32 which makes the answer 31 but that is not even given in answer choices and the answer they say should be 27 . please help
We want to maximize x and minimize y, and this is:

$\displaystyle (x,y)=(3,-7)$

And so:

$\displaystyle 2x-3y=6+21=27$
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November 12th, 2018, 09:55 AM   #3
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Let $\displaystyle 2x-3y$ be a sum $\displaystyle 2x-3y=2x+(-3y)$
Maximize of $\displaystyle -3y$ means minimize of $\displaystyle 3y$

Last edited by idontknow; November 12th, 2018 at 10:11 AM.
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November 12th, 2018, 11:54 AM   #4
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Quote:
Originally Posted by ketanco View Post
x and y integer
-5<x<4
-8<y<1

what is the maximum value of 2x-3y?

when i multiply first eqn by 2 and second by -3 and the add them up i get
-13<2x-3y<32 which makes the answer 31 but that is not even given in answer choices and the answer they say should be 27 . please help
When you deal with inequalities involving integers make them $\le$ rather than $<$.

$-\ 5 < x < 4 \text { and } x \in \mathbb Z \implies -\ 4 \le x \le 3 \implies -\ 8 \le 2x \le 6.$

$-\ 8 < y < 1 \text { and } y \in \mathbb Z \implies -\ 7 \le y \le 0 \implies 21 \ge -\ 3y \ge 0 \implies 0 \le -\ 3y \le 21.$

$\therefore -\ 8 + 0 \le 2x - 3y \le 6 + 21 \implies -\ 8 \le 2x - 3y \le 27.$
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November 12th, 2018, 01:03 PM   #5
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What about maximize $\displaystyle \frac{x}{y}$ ?
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