Math Problem Statement
Graph the system of inequalities and find three points that are solutions to the system.
Solution
The system of inequalities provided is:
y > 4x - 1 \\ y < \frac{1}{4}x - 2 \end{cases}$$ ### Analysis of the Graph: - **First inequality**: $$y > 4x - 1$$ - The boundary line is $$y = 4x - 1$$, which is represented by the **dashed red line**. The shading above this line indicates that the inequality includes values where $$y$$ is greater than the boundary. - **Second inequality**: $$y < \frac{1}{4}x - 2$$ - The boundary line is $$y = \frac{1}{4}x - 2$$, which is represented by the **dashed blue line**. The shading below this line indicates that the inequality includes values where $$y$$ is less than the boundary. ### Solution Region: The overlapping region between the red and blue shading is the solution to this system of inequalities. Based on the graph, this region is a triangular section located near the negative x-values and negative y-values. ### Part 2: Coordinates of Three Points in the Solution Region Three points that lie in the overlapping region (the solution to both inequalities) are: 1. $$(-2, -4)$$ 2. $$(-3, -5)$$ 3. $$(-4, -6)$$ These points are located in the dark purple region, where both inequalities hold true. --- Would you like more details on how the inequalities were graphed or how to solve similar systems? ### Five Related Questions: 1. How do you determine which side of the line to shade for an inequality? 2. What does the dashed line represent in a graph of inequalities? 3. How can you solve a system of inequalities algebraically instead of graphically? 4. How do you know when a point is a solution to a system of inequalities? 5. What would happen if one of the inequalities had a "greater than or equal to" sign? ### Tip: When graphing inequalities, always check a point (like the origin) to confirm which side of the boundary line should be shaded.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Inequalities
Graphing
Systems of Inequalities
Formulas
y > 4x - 1
y < (1/4)x - 2
Theorems
Linear Inequalities
System of Inequalities
Suitable Grade Level
Grades 9-11