How do you solve the system of inequalities #2x + 5y \leq - 20# and #- 5x + 4y > - 4#?

It is that region that is below or on #y=-2/5x-4# and at the same time above the plot #y=5/4x-1#. All as the shaded region on the graph.

Explanation:

It is that condition which satisfies both conditions.

Note that #-5x+4y > -4# can never take on the value of -4 so by convention this condition is shown as a dotted line.

#color(blue)("Consider "2x+5y <=-20)#

Solid line as it may take on the value of -20

To plot this line we manipulate as follows:

Subtract #2x# from both sides

#5y<=-2x-20#

Divide both sides by 5

#y<=-2/5x-4 #

The feasible region for this:
Select any value #x#. Draw a vertical line through it. #y# may take on any value on that vertical line that is either on the plot of #y=-2/5x-4# or below it.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Consider "-5x+4y > -4)#

Dotted line as it may not take on the value of -4

To plot this we manipulate as follows:

Add #5x# to both sides

#4y > 5x-4#

Divide both sides by 4

#y > 5/4x-1#

The feasible region for this:

Select any value for #x#. Draw a vertical line through it. #y# may take on any value that is above the plot #y=5/4x-1#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Consider the combined feasible regions")#

It is that region that is below or on #y=-2/5x-4# and at the same time above the plot #y=5/4x-1#. All as the shaded region on the graph.