Which Graph Represents the Solution Set of the System of Inequalities

Testing your understanding of linear inequalities can be approached in several ways, from the traditional method of graphing them yourself to the more strategic approach of using process of elimination and logical deduction. This article aims to guide you through solving a system of inequalities and identifying the correct graph among the answer choices. Let's dive into the process step-by-step.

The System of Inequalities

The given system of inequalities is:

y ≤ 2x 1 y > -2x - 3

These inequalities represent two lines on a coordinate plane and the regions defined by these lines. By graphing these inequalities, we can determine their solution set, which is the overlapping area where both inequalities are satisfied.

Graphing the Inequalities

Step 1: Graphing the First Inequality y ≤ 2x 1

The graph is linear, represented by a straight line. The inequality is ≤ (less than or equal to), meaning the line will be solid. The shading will be below the line, indicating all points (y, x) that satisfy y ≤ 2x 1. The slope of the line is 2, which means it is steep and slopes upward. The y-intercept is 1, meaning the line crosses the y-axis at the point (0, 1).

Plot the y-intercept at (0, 1) and use the slope to draw the line, which rises 2 units for every 1 unit it moves to the right (or down 2 units for every 1 unit it moves to the left).

Step 2: Graphing the Second Inequality y > -2x - 3

The graph is also linear, represented by a straight line. The inequality is > (greater than), meaning the line will be dashed. The shading will be above the line, indicating all points (y, x) that satisfy y > -2x - 3. The slope of the line is -2, which means it is steep and slopes downward. The y-intercept is -3, meaning the line crosses the y-axis at the point (0, -3).

Plot the y-intercept at (0, -3) and use the slope to draw the line, which falls 2 units for every 1 unit it moves to the right (or rises 2 units for every 1 unit it moves to the left).

Identifying the Solution Set

The solution to the system of inequalities is the area where the shaded regions of both inequalities overlap. This overlapping area represents all (x, y) points that satisfy both inequalities simultaneously.

Evaluating the Answer Choices

With both inequalities graphed, let's review the four possible answer choices:

Graph A Graph B Graph C Graph D

Upon careful inspection, you should be able to identify which graph correctly represents the solution set. The correct graph will show a solid line for y ≤ 2x 1, a dashed line for y > -2x - 3, and the overlapping shaded region where both inequalities are satisfied.

Given the analysis, the correct answer is Graph A.

Conclusion

Solving systems of inequalities through graphing or logical elimination can be a powerful tool in your mathematical arsenal. By understanding the properties of linear equations and inequalities, you can readily determine the solution set and identify the correct graph among multiple answer choices. This skill is particularly useful in standardized tests and real-world applications where analyzing data and understanding relationships between variables is crucial.