Which system of linear inequalities is represented by the graph? y > x – 2 and x – 2y < 4 y > x + 2 and x + 2y < 4 y > x – 2 and x + 2y < 4 y > x – 2 and x + 2y < –4
The Correct Answer and Explanation is:
To determine the system of linear inequalities represented by a graph, we need to analyze the features of the graph and match it with the appropriate system of inequalities. Let’s break down each of the inequalities one by one and then analyze the graph:
Step 1: Analyze the inequalities
- y > x – 2:
- This inequality represents a region above the line (y = x – 2).
- The line (y = x – 2) has a slope of 1 and a y-intercept of -2.
- Since the inequality is strict ((>)), the line itself is dashed (not included in the solution set).
- x – 2y < 4:
- Rearranging this inequality, we get:
[
x – 2y < 4 \implies 2y > x – 4 \implies y > \frac{x – 4}{2}
] - This inequality represents a region above the line (y = \frac{x – 4}{2}), which has a slope of 1/2 and a y-intercept of -2.
- Since the inequality is strict ((<)), the line itself is also dashed.
Step 2: Interpret the graph
Now, let’s evaluate the graph based on the inequalities.
- y > x – 2 indicates a region above the line (y = x – 2). The line should be dashed, meaning it is not part of the solution.
- x – 2y < 4 simplifies to (y > \frac{x – 4}{2}), which means the graph will be shaded above the line (y = \frac{x – 4}{2}).
Step 3: Check each system of inequalities
- y > x – 2 and x – 2y < 4: This system matches our interpretation. The graph should show a dashed line for (y = x – 2) and a dashed line for (y = \frac{x – 4}{2}), with the solution region being the area above both lines.
- y > x + 2 and x + 2y < 4: This system involves different lines and would create a different shading region, not matching the graph.
- y > x – 2 and x + 2y < 4: While this system has one correct line (y > x – 2), the second inequality would represent a different line and shading, so it does not match the graph.
- y > x – 2 and x + 2y < -4: This system has a similar line (y > x – 2), but the second inequality would create a region below a line, not above, so this does not match the graph.
Conclusion
The correct system of inequalities represented by the graph is y > x – 2 and x – 2y < 4. This system creates the proper shading above both the lines, consistent with the given graph.