Math Problem Statement
Graph the given systems of inequalities on graph paper and provide three solutions for each.
Solution
The problem involves graphing systems of inequalities and finding solutions for each system. Here's the breakdown of each system of inequalities:
System 1
x + y &\leq 5, \\ x + 2y &< -3. \end{aligned}$$ - Graph both inequalities: 1. Rewrite $$x + y \leq 5$$ as $$y \leq -x + 5$$. 2. Rewrite $$x + 2y < -3$$ as $$y < -\frac{1}{2}x - \frac{3}{2}$$. - The solution region is the intersection of the shaded areas of these two inequalities. --- ### **System 2** $$\begin{aligned} y &\geq 4x + 1, \\ y &\leq -x + 2. \end{aligned}$$ - Graph both inequalities: 1. The inequality $$y \geq 4x + 1$$ represents a region above the line $$y = 4x + 1$$. 2. The inequality $$y \leq -x + 2$$ represents a region below the line $$y = -x + 2$$. - The solution is the region where these areas overlap. --- ### **System 3** $$\begin{aligned} y &< -3x + 4, \\ y &> -\frac{4}{3}x + 2. \end{aligned}$$ - Graph both inequalities: 1. The inequality $$y < -3x + 4$$ is the area below the line $$y = -3x + 4$$. 2. The inequality $$y > -\frac{4}{3}x + 2$$ is the area above the line $$y = -\frac{4}{3}x + 2$$. - The solution is the intersection of these regions. --- ### Solutions: For each system, three solutions are points that fall within the overlapping shaded region. 1. **System 1**: Test points such as $$(-4, 1)$$, $$(-5, 0)$$, $$(-3, -1)$$. 2. **System 2**: Test points such as $$(0, 2)$$, $$(-1, 3)$$, $$(0.5, 3)$$. 3. **System 3**: Test points such as $$(-1, 4)$$, $$(1, -1)$$, $$(0, 3)$$. Would you like me to graph these for visual clarity or provide a detailed explanation for solving step by step? --- ### Follow-Up Questions: 1. How do you graph inequalities involving less than or equal to ($$\leq$$) versus strict inequality ($$<$$)? 2. What are the key differences in solving systems of inequalities compared to systems of equations? 3. Can you explain how to test if a point lies in the solution region for a system of inequalities? 4. How would you describe the meaning of the intersection region in inequality graphs? 5. Can inequalities ever have no overlapping region, and how would you identify that? ### **Tip:** Always use test points to verify the solution region, especially if the inequality's shading isn't clear.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Inequalities
Graphing Linear Inequalities
Intersection of Regions
Formulas
y = mx + b for converting inequalities into slope-intercept form
Testing points to verify solutions within the solution region
Theorems
Intersection of regions theorem: The solution set of a system of inequalities is the overlap of the solution sets of each individual inequality.
Suitable Grade Level
Grades 9-12