We draw the lines

y=2x-8, y=10-x, y=, x=1, and y=2 

 

Maximization 
Maximize p = 7x - 4y subject to
y ≧ 2x - 8,  the red line y=2x-8 and the region above it, since it's y≧
y ≦ 10 - x, the green line y=10-x and the region below it, since it's y≦
y ≦ , the blue line y= and the region below it, since it's y≦
x ≧ 1, the black line x=1 and the region to the right of it, since it's x≧
y ≧ 2, the purple line y=2 and the region above it, since it's y≧


We will chop off the unnecessary parts of the lines, leaving 
only the region bounded by them, which is this pentagon:

 

Now we must find all the corner points of this pentagon, by solving these
five systems:

 the red and green lines, which gives (6,4)
 the red and purple lines, which gives (5,2)
 the black and purple lines, which gives (1,2)
 the blue and black lines, which gives (1,4.5).
 the blue and green lines, which gives (4,6)


 
We evaluate p = 7x - 4y at each of those corner points:

Corner point | p = 7x - 4y
    (1,2)    |     -1
    (5,2)    |     27
    (6,4)    |     26
    (4,6)    |      4
  (1,4.5)    |    -11

So the maximum value of p=27 occurs when x=5 and y=2

Edwin