x ≥ 0, y ≥ 0, 2 x + y &lt; 8 x + 2 y &lt; 10Does x + y have a maximum value under the above conditions?

Question

x  ≥  0,  y  ≥  0,  2  x  +  y  &amp;lt;  8  x  +  2  y  &amp;lt;  10Does   x  +  y have a maximum value under the above conditions?

treccinair · Accepted Answer

The two latter inequalities imply   x  +  y  &amp;lt;  6 so   x  +  y can not equal six. On the other hand, the point   x  =  2  ,  y  =  4 belongs to the boundary of the region. Therefore,   x  +  y being a continous function can be arbitrarily close to   6, hence the answer is negative.

sweetymoeyz · Accepted Answer

Assuming that you are looking for integers:
First you need to remove the strict inequalities.
In order for

2 x + y < 8

to be true, where x and y are integers, it must the case that

2 x + y \leq 7

In order for

x + 2 y < 10

to be true, where x and y are integers, it must the case that

x + 2 y \leq 9

This system has maxima in the integers.

x ≥ 0 , y ≥ 0 , 2 x + y < 8 x

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