Question

How do you graph the system of linear inequalities `y>=-4` and `y<-2x+10`?

Answer 1

See below

Explanation:

As for the first inequality, `y \ge -4` represents all the points whose `y` coordinate is greater than `-4`, no matter what their `x` coordinate is. So, the inequality is satisfied by the following infinite area:

graph{y>=-4 [-10, 10, -7, 2]}

On the other hand, `y<-2x+10` is represented by all the points below the line `y = -2x+10`. In fact, all the points on the line are such that their `y` component is exactly `-2x+10` If we choose any point below the line, the `y` component will be less than `-2x+10`, which is exactly what the inequality asks.

To draw the line, simply find two of its points by choosing two random `x` values:

`x = 0 \implies y = -2\cdot 0 + 10 = 10 \to P_1 = (0,10)`
`x = 5 \implies y = -2\cdot 5 + 10 = 0 \to P_2 = (5,0)`

Here's the graph:

graph{y<-2x+10 [-10, 10, -7, 2]}

Now, assume that you drew the inequality graphs on the same plane. Let's see that you coloured with blue the points satisfying the first inequality, and in red the points satisfying the second.

The points that solve the system are those who satisfy both equations simultaneously, and thus are red and blue at the same time: you're looking for violet points! :)