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.
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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