We can represent a set of linear inequalities on a graph.

The method used is very similar to when solving equations.

You first need to plot the equation, as if the inequality sign was an equals sign.

For example:

Shade in the region that is valid for

1. y > x - 3     [red]


2. y < 0           [green]

3. -3 < x < 3   [blue]

Firstly, we draw all 3 lines. 

1. y = x - 3     [red]


2. y = 0           [green]

3. x = -3 and x = 3 [blue]


Now, we consider the inequalities

1. y > x - 3 (red line) is asking us when is y greater than (x - 3), so we don't want any values where y would be less than this.

This can be quite confusing. A useful tip is to pick a point from each region (here we only have two regions. Let's call this left and right region to make it easier. 

Suppose we pick a random point (4,-2) (which belongs to the right region). Check whether this fits the criteria.

y = -2
x = 4
x - 3 = 4 - 3 = 1 > -2 = y

So, no, this point is not valid, which means this whole region is not valid. 

To double check, pick a point on the left region (1,1).

y = 1
x = 1
x - 3 = 1 - 3 = -2 < 1 = y

Yes, it does work! Which is what we were looking for.

This means we can shade in the right region, because we don't want any values from there.

2. y < 0 

We only want to consider negative values of y, so clearly we shade in the top region - we don't want any values from there.

3. -3 < x < 3

We only want values between -3 and 3. So we shade in anything smaller than x = -3 (the left hand side of the x=-3 line) and we shade in anything bigger than x=3 (the right hand side of x=3).

We should now be left with a triangular region in the middle, that is not shaded in on the graph.


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