Which two quadrants contain all of the solutions to the following system?
y>4x-2
y>-3x+5
A. I and II B. II and III C. III and IV D. I and IV
D. I and IV
To determine which two quadrants contain all of the solutions to the system of inequalities, we need to find the regions where both inequalities are simultaneously true.
First, let's analyze the inequality y > 4x - 2. This inequality has a positive slope of 4 and a y-intercept of -2. To graph it, we can start by plotting the y-intercept at (0, -2) and then use the slope to find additional points. For example, if we increase x by 1, y should increase by 4, giving us the point (1, 2). After connecting these two points with a dashed line, we shade the region above the line to represent the inequality.
Next, let's analyze the inequality y > -3x + 5. This inequality has a negative slope of -3 and a y-intercept of 5. Similarly, we can plot the y-intercept at (0, 5) and use the slope to find additional points. For example, if we decrease x by 1, y should increase by 3, giving us the point (-1, 8). After connecting these two points with a dashed line, we shade the region above the line to represent the inequality.
Now, let's identify the region that satisfies both inequalities. Since both inequalities have the ">" symbol, we need to consider the area above both dashed lines. Looking at the graph, we can see that the regions above the lines (including the lines themselves) intersect in Quadrant I (top right) and Quadrant IV (bottom right).
Therefore, the two quadrants that contain all of the solutions to the system are Quadrants I and IV. So, the correct answer is option D.
To determine which two quadrants contain all of the solutions, we need to find the regions where both inequalities are satisfied.
Let's start by solving the first inequality:
y > 4x - 2
To graph this inequality, we can start by drawing the line y = 4x - 2 (which is the boundary line). Since it is a "greater than" inequality, we'll draw the line as a dashed line:
y = 4x - 2
Next, we need to determine which side of the line satisfies the inequality. Since it is y > 4x - 2, we shade the region above the line:
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Now let's solve the second inequality:
y > -3x + 5
Again, we start by drawing the line y = -3x + 5 (boundary line):
y = -3x + 5
Since it is a "greater than" inequality, we draw the line as a dashed line. Next, we shade the region above the line (since it is y > -3x + 5):
```
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```
Now, we need to find the region where both inequalities are satisfied. It is the region where the shaded areas from both inequalities overlap.
This region is Quadrant I, since it is the area where both inequalities are satisfied:
```
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```
Therefore, the two quadrants that contain all of the solutions to the system of inequalities are Quadrant I and Quadrant II.
Thus, the answer is A. I and II.