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Graphing a single inequality involves two main steps.
We can tell a lot of information about the boundary lines from the inequalities given in the system.
Let's find each of these key pieces of information for the inequalities in the system.
Information | Inequality (I) | Inequality (II) |
---|---|---|
Given Inequality | 2x+6y>12 | 3x+9y≤27 |
Boundary Line Equation | 2x+6y=12 | 3x+9y=27 |
Solid or Dashed? | > ⇒ Dashed | ≤ ⇒ Solid |
y=mx+b | y=-31x+2 | y=-31x+3 |
Great! With all of this information, we can plot the boundary lines.
Before we can shade the solution set for each inequality, we need to determine on which side of the plane their solution sets lie. To do that, we will need a test point that does not lie on either boundary line.
It looks like the point (0,0) would be a good test point. We will substitute this point for x and y in the given inequalities and simplify. If the substitution creates a true statement, we shade the same region as the test point. Otherwise, we shade the opposite region.
Information | Inequality (I) | Inequality (II) |
---|---|---|
Given Inequality | 2x+6y>12 | 3x+9y≤27 |
Substitute (0,0) | 2(0)+6(0)>?12 | 3(0)+9(0)≤?27 |
Simplify | 0≯12 | 0≤27 |
Shaded Region | opposite | same |
For Inequality (I) we will shade the region opposite our test point, or above the boundary line. For Inequality (II), however, we will shade the region containing the test point, or below the boundary line.
Finally, we can view only the solution set by removing the shaded regions that are not overlapping.