Question

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

Answer 1

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.

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`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`
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`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.