Math Problem Statement

Aubrey graphed the following system of linear inequalities. Which graph best represents the solution set to this system of inequalities? The system is: y < -x + 1 and x - y ≤ 2.

Solution

To solve this problem, let’s examine each inequality separately and how they affect the graph.

Step 1: Analyzing the Inequalities

  1. First Inequality: y<x+1y < -x + 1

    • This inequality has a boundary line y=x+1y = -x + 1, which has a slope of -1 and a y-intercept of 1.
    • Since it’s a "<" inequality (strict inequality), the line y=x+1y = -x + 1 should be dashed to indicate that points on the line are not included in the solution.
    • The shading for this inequality will be below the line, as yy values less than x+1-x + 1 satisfy the inequality.
  2. Second Inequality: xy2x - y \leq 2

    • Rearranging this inequality, we get yx2y \geq x - 2.
    • The boundary line for this inequality is y=x2y = x - 2, with a slope of 1 and a y-intercept of -2.
    • Since it’s a "≤" inequality (inclusive inequality), the line y=x2y = x - 2 should be solid, indicating that points on the line are included in the solution.
    • The shading for this inequality will be above the line, as yy values greater than or equal to x2x - 2 satisfy the inequality.

Step 2: Identifying the Correct Graph

Now, we need to find a graph where:

  • The region below the dashed line y=x+1y = -x + 1 is shaded.
  • The region above the solid line y=x2y = x - 2 is shaded.
  • The solution set (overlap of the shaded regions) is correctly represented.

Conclusion

Looking at the options provided:

  • Option B is the correct graph, as it shows the shaded region below the dashed line y=x+1y = -x + 1 and above the solid line y=x2y = x - 2.

Thus, the answer is (B).

Would you like a more detailed breakdown of each step or have any further questions?

Related Questions

  1. How would the solution region change if both inequalities were strict inequalities?
  2. How does shading below or above the line affect the solution region in a graph?
  3. What is the effect of changing the slope on the boundary line of a linear inequality?
  4. How do solid and dashed lines help differentiate between inclusive and strict inequalities?
  5. Can you describe the intersection of solution regions for systems of inequalities in different quadrants?

Tip

When graphing inequalities, remember that dashed lines represent strict inequalities (without equality), while solid lines indicate inclusive inequalities (with equality). This helps clearly indicate whether points on the line are part of the solution.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Inequalities
Graphing Inequalities

Formulas

y = -x + 1 (Boundary Line for y < -x + 1)
y = x - 2 (Boundary Line for x - y ≤ 2, rearranged)

Theorems

Intersection of Inequalities Solution Regions
Concept of Solid and Dashed Lines in Graphing

Suitable Grade Level

Grades 8-10