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Graph each inequality separately. The solution will be the intersection, or overlap, of the shaded regions.
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 | 8x+4y≥10 | 3x−6y>12 |
Boundary Line Equation | 8x+4y=10 | 3x−6y=12 |
Solid or Dashed? | ≥ ⇒ Solid | > ⇒ Dashed |
y=mx+b | y=-2x+25 | y=21x+(-2) |
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.
We will use (0,0) as a test point. To do so, we will substitute 0 for x and y in the given inequalities and simplify. If the substitution creates a true statement, we will shade the region that contains the point (0,0). Otherwise, we will shade the region that does not contain the point.
Information | Inequality (I) | Inequality (II) |
---|---|---|
Given Inequality | 8x+4y≥10 | 3x−6y>12 |
Substitute (0,0) | 8(0)+4(0)≥?10 | 3(0)−6(0)>?12 |
Simplify | 0≱10 | 0≯12 |
Shaded Region | does not contain (0,0) | does not contain (0,0) |
For Inequality (I), we will shade the region opposite our test point, or above the boundary line. For Inequality (II) we will also shade the region opposite the test point. This time, however, it will be below the boundary line.
The solution set of our system is the intersection of both shaded regions.