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Graph each inequality. The solution will be the intersection, or overlap, of the shaded regions.
Graphing a single inequality involves two main steps.
For this exercise, we need to do this process for each of the inequalities in the system. y>2x+4 & (I) 2x-y≤ 4 & (II) The system's solution set will be the intersection of the shaded regions in the graphs of Inequalities (I) and (II).
Let's find each of these key pieces of information for the inequalities in the system.
Information | Inequality (I) | Inequality (II) |
---|---|---|
Given Inequality | y >2x+4 | 2x-y ≤4 |
Boundary Line Equation | y =2x+4 | 2x-y =4 |
Solid or Dashed? | > ⇒ Dashed | ≤ ⇒ Solid |
y= mx+ b | y= 2x+ 4 | y= 2x+( - 4) |
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 shade the region which does not contain the point.
Information | Inequality (I) | Inequality (II) |
---|---|---|
Given Inequality | y>2x+4 | 2x-y≤ 4 |
Substitute ( 0, 0) | 0? >2( 0)+4 | 2( 0)- 0? ≤4 |
Simplify | 0≯ 4 | 0≤ 4 |
Shaded Region | does not contain (0,0) | contains (0,0) |
For Inequality (I) we will shade the region opposite to the test point, or above the boundary line. For Inequality (II), however, we will shade the region containing our test point, or above the boundary line. The solution set of the system is the intersection of both shaded regions.