Math Problem Statement

Which graph represents the solution set of the system of inequalities {y < -3x - 2, y ≤ x - 2}?

Solution

Let's analyze the system of inequalities given in the problem:

y < -3x - 2 \\ y \leq x - 2 \end{cases}$$ 1. **Graphing the first inequality $$y < -3x - 2$$:** - The boundary line is $$y = -3x - 2$$, which is a dashed line because the inequality is strict ($$<$$). - Slope = $$-3$$ (line goes down 3 units for every 1 unit right), and y-intercept = $$-2$$ (the line crosses the y-axis at $$-2$$). - The solution region is below this line. 2. **Graphing the second inequality $$y \leq x - 2$$:** - The boundary line is $$y = x - 2$$, which is a solid line because the inequality includes equality ($$\leq$$). - Slope = $$1$$ (line goes up 1 unit for every 1 unit right), and y-intercept = $$-2$$. - The solution region is below or on this line. 3. **Solution Region:** - The solution is the overlapping shaded region of the two inequalities: below the dashed line $$y = -3x - 2$$ and below/on the solid line $$y = x - 2$$. ### Correct Graph: The graph representing this solution set has: - A dashed line for $$y = -3x - 2$$ with shading below it. - A solid line for $$y = x - 2$$ with shading below or on it. - The overlapping region forms a triangular area. **Answer:** The correct graph is **bottom-left (third graph)**. Would you like me to explain how these inequalities are plotted in more detail? --- ### Further Questions to Explore: 1. What is the significance of a solid line versus a dashed line in graphing inequalities? 2. How do we determine the slope and intercept of a line from an equation? 3. What happens if the inequality directions ($$<$$ or $$\leq$$) are flipped? 4. How would the solution region change if one inequality was flipped to $$>$$? 5. How do we confirm the solution region by testing points? --- ### Tip: When solving systems of inequalities, always test a point (like $$(0, 0)$$) to confirm the shaded region is correct.

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Graphing Inequalities
Systems of Inequalities

Formulas

Slope-intercept form of a line: y = mx + b

Theorems

Intersection of solution regions for inequalities

Suitable Grade Level

Grades 8-10

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