<PRE><B>Question 1148055</B>
.
&lt;pre&gt;
Let me, first, to rewrite your constraints in equivalent form


    y &lt;=  2 + {{{x/3}}}    (1)    (first  given constraint)

    y &gt;= -2 + {{{x/3}}}    (2)    (second given constraint) 

    y &gt;=  6 - x    (3)     (third  given constraint) 


The plot is shown below.


    {{{graph( 330, 330, -10, 10, -10, 10,
          (1/3)x +2, (1/3)x-2, 6-x
)}}}


    Plot  x-3y = -6  (red),  x-3y = 6  (green) and  x+y = 6 (blue).


So, your feasible area is the set of points between two parallel lines (red and green; constraint (1) and (2)), that are ABOVE blue line.


The given linear objective function  z = x + 3y,  &lt;U&gt;OBVIOUSLY, HAS NO maximum&lt;/U&gt;

    (its value can be made arbitrary large).


&lt;U&gt;It has a minimum value, instead&lt;/U&gt;.


This minimum value is its value in one of the two intersection (corner) points.


The intersection points are  (3,3)  and (6,0).


The values of the objective function  z = x + 3y  at these points are


    at (3,3)  z = 3 + 3*3 = 12

    at (0,6)  z = 6 + 3*0 = 6.


&lt;U&gt;ANSWER&lt;/U&gt;.  The given objective function HAS NO maximum value.

         It has minimum value at the point (x,y) = (6,0).
&lt;/pre&gt;


Solved, answered, explained and completed.


--------------------


To see many other similar problems solved by the Linear Programming method and to learn how the method works, &amp;nbsp;look into the lesson

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-minimax-problems-by--the-Linear-Programming-method.lesson&gt;Solving minimax problems by the Linear Programming method&lt;/A&gt; 

in this site.


Also, &amp;nbsp;look into the solutions of other similar problems in the archive to this forum under the links


https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1142000.html 

https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1137172.html

https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1136382.html

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