Question

What is an easy way to find points satisfying `(8x+4y)/32 > 1` ?

Answer 1

See below

Explanation:

`(8x+4y)/32 > 1`

[N.B. We are asked for an "easy way" to find points satisfying the above. Whether or not the following is considered "easy" is entirely subjective. It is the way I would approach the problem.]

Let's consider the limiting case where:

`(8x+4y)/32 = 1`

This is a simple linear function as follows.

`(8x+4y) = 32`

`4y = -8x+32`

`y = 1/4(-8x+32) = -2x+8`

So now we can graph the limiting line as follows:

graph{-2x+8 [-10, 10, -5, 5]}

Now we need to consider the inequality.

`8x+4y >32`

`4y > -8x+32 -> y > -2x+8`

`:. y` is the set of all points greater but not equal to our limiting line above. This area is shaded below.

graph{(8x+4y)/32 > 1 [-10, 10, -5, 5]}