Math Problem Statement
solve
Solution
To solve this problem, let's analyze the given system of inequalities and determine which graph correctly represents the solution set.
The system of inequalities is:
1.\ & y + \frac{3}{2}x < 2 \\ 2.\ & y - 1 \geq 2x \end{aligned}$$ ### Step 1: Rearrange the inequalities to slope-intercept form (y = mx + b) 1. **First Inequality:** $$y + \frac{3}{2}x < 2 \quad \Rightarrow \quad y < -\frac{3}{2}x + 2$$ The boundary line is $$y = -\frac{3}{2}x + 2$$, and the region below this line (since the inequality is "<") is shaded. 2. **Second Inequality:** $$y - 1 \geq 2x \quad \Rightarrow \quad y \geq 2x + 1$$ The boundary line is $$y = 2x + 1$$, and the region above this line (since the inequality is "≥") is shaded. ### Step 2: Identify the correct graph - The correct graph will have the region below the line $$y = -\frac{3}{2}x + 2$$ shaded and the region above the line $$y = 2x + 1$$ shaded. - The solution set is where these two shaded regions overlap. ### Analysis of the Graphs: - Look for the graph where the overlapping region is in the correct location according to the inequalities. Upon examining the provided graphs: - The graph marked with an asterisk (*) correctly shows the overlapping shaded region, where: - The area below the line $$y = -\frac{3}{2}x + 2$$ and above the line $$y = 2x + 1$$ is shaded. ### Conclusion: The graph with the asterisk (*) is the correct representation of the system of inequalities. Would you like more details on how to interpret the inequalities, or do you have any other questions? ### Follow-Up Questions: 1. How do you determine the slope of a line from an equation? 2. What does the inequality symbol "<" or ">" indicate in terms of shading on a graph? 3. How would the graph change if one of the inequalities was non-strict (e.g., "≤" or "≥")? 4. What is the significance of the point where the two lines intersect? 5. Can a system of inequalities have no solution, and what would that look like on a graph? 6. How does the slope-intercept form help in graphing linear inequalities? 7. What happens if the inequalities were strict (e.g., "<" and ">")? How would that affect the graph? 8. How do you check if a specific point is a solution to a system of inequalities? **Tip:** When graphing systems of inequalities, always pay attention to the direction of the inequality sign to correctly shade the regions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Inequalities
Graphing Inequalities
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12