Solve a set of linear inequalities in 2 variables by shading a region of a graph
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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