How to graph an interval of real numbers? 
Assume that intergers $m$ and $n$ satisfy $2|m|+3|n-1|\leq 7$. $m+n$ is maximum when $(m,n) = (3,?), (?,?)$ and its maximum value is $?$

given the above question the first thing i tried was calculating the intervals for each variable like this:
$2|m| \leq 7 \therefore m \in [-\frac{7}{2},\frac{7}{2}]$
$3|n-1| \leq 7 \therefore n \in [-\frac{4}{3},\frac{10}{3}]$
and since the exercise is telling me that the maximum is reached when $m=3$
$2|3|+3|n-1|\leq 7 \therefore 7 \geq 3|n-1| + 6 \therefore n \in [-\frac{1}{3},\frac{8}{3}]$
The next thing i want to do is make a graph but i got stuck in this part since i newbie in linear programming, any advice or material in this topic?
 A: 
Assume that integers $m$ and $n$ satisfy $2|m|+3|n−1|≤7$. $m+n$ is maximum when $(m,n)=(3,?),(?,?)$ and its maximum value is $?$

We can turn the following problem into an LP model like such:
$$\max z = m+n$$
$$s.t.\quad 2m \le 7$$
$$2m \ge -7$$
$$3(n-1)\le7$$
$$3(n-1)\ge-7$$
$$m,n\in\Bbb Z$$
The linearly relaxed, feasible region of this model will look like so:

Then, we can add in all the integer points that exist within the feasible region like so (which is the real feasible region of the above model):

Since we know $m=3$, we can isolate all the points where that is true:

Which are the following points:
$$\left(3,-1\right),\left(3,0\right),\left(3,1\right),\left(3,2\right),\left(3,3\right)$$
From here, since we’re looking for a point to satisfy the original $2||+3|−1|≤7 $ constraint, we can conclude that biggest, valid point that maximizes $z=m+n$ is the point $\left(3,1\right)$.
A: I have renamed your variables as $x$ and $y$ to match the linked Desmos plot.
Introduce (nonnegative) variables $u$ and $v$ to represent the absolute values.  The resulting linear problem is to maximize $m+n$ subject to
\begin{align}
2u+3v &\le 7 \\
u &\ge x \\
u &\ge -x \\
v &\ge y-1 \\
v &\ge -(y-1) \\
x &\in \mathbb{Z} \\
y &\in \mathbb{Z}
\end{align}
The two optimal solutions turn out to be $(x,y,u,v)=(3,1,3,1)$ and $(x,y,u,v)=(2,2,2,2)$.
See https://www.desmos.com/calculator/x2w97ng3up
