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Sketch the region that corresponds to the given inequalities. 4x + 5y s 4 4x - 5y 2 4 -y 2 2

Sketch the region that corresponds to the given inequalities. 4x + 5y s 4 4x - 5y 2 4 -y 2 2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Title:** Understanding Linear Inequalities and Solution Sets **Objective:** Sketch the region that corresponds to the given inequalities: - \(4x + 5y \leq 4\) - \(4x - 5y \geq 4\) - \(-y \geq 2\) ### Explanation of Graphs The provided images illustrate the graphical representation of the solution set for the three inequalities. Each graph shows the \(x\)- and \(y\)-coordinate plane with specific shaded regions representing where the inequalities hold true. 1. **Top Left Graph:** - Displays the solution for the inequality \(4x + 5y \leq 4\). - The shaded region is below the line \(4x + 5y = 4\). - The "Solution Set" is the area on this graph where the shading occurs, meaning all points in this region satisfy the inequality. 2. **Top Middle Graph:** - Represents the inequality \(4x - 5y \geq 4\). - The shaded region is above the line \(4x - 5y = 4\). - The "Solution Set" includes the shaded area where points satisfy this inequality. 3. **Top Right Graph:** - Shows the simultaneous solution for all the given inequalities. - Includes \(4x + 5y \leq 4\), \(4x - 5y \geq 4\), and \(-y \geq 2\). - The unshaded triangular region is the intersection of the solution sets for the individual inequalities. - This region contains points that satisfy all three inequalities concurrently. 4. **Bottom Left Graph:** - Focuses on the inequality \(-y \geq 2\) which simplifies to \(y \leq -2\). - The shaded area is below the line \(y = -2\), indicating where the inequality holds true. - The "Solution Set" here marks the area that satisfies this specific condition. **Conclusion:** The graphs collectively demonstrate the process of finding and visualizing the solution region for systems of linear inequalities. The graphs guide students in understanding how to determine which regions on a coordinate plane satisfy all the inequalities simultaneously, emphasizing the concept of the intersection of solution sets.
Transcribed Image Text:**Title:** Understanding Linear Inequalities and Solution Sets **Objective:** Sketch the region that corresponds to the given inequalities: - \(4x + 5y \leq 4\) - \(4x - 5y \geq 4\) - \(-y \geq 2\) ### Explanation of Graphs The provided images illustrate the graphical representation of the solution set for the three inequalities. Each graph shows the \(x\)- and \(y\)-coordinate plane with specific shaded regions representing where the inequalities hold true. 1. **Top Left Graph:** - Displays the solution for the inequality \(4x + 5y \leq 4\). - The shaded region is below the line \(4x + 5y = 4\). - The "Solution Set" is the area on this graph where the shading occurs, meaning all points in this region satisfy the inequality. 2. **Top Middle Graph:** - Represents the inequality \(4x - 5y \geq 4\). - The shaded region is above the line \(4x - 5y = 4\). - The "Solution Set" includes the shaded area where points satisfy this inequality. 3. **Top Right Graph:** - Shows the simultaneous solution for all the given inequalities. - Includes \(4x + 5y \leq 4\), \(4x - 5y \geq 4\), and \(-y \geq 2\). - The unshaded triangular region is the intersection of the solution sets for the individual inequalities. - This region contains points that satisfy all three inequalities concurrently. 4. **Bottom Left Graph:** - Focuses on the inequality \(-y \geq 2\) which simplifies to \(y \leq -2\). - The shaded area is below the line \(y = -2\), indicating where the inequality holds true. - The "Solution Set" here marks the area that satisfies this specific condition. **Conclusion:** The graphs collectively demonstrate the process of finding and visualizing the solution region for systems of linear inequalities. The graphs guide students in understanding how to determine which regions on a coordinate plane satisfy all the inequalities simultaneously, emphasizing the concept of the intersection of solution sets.
**Problem Statement:** Determine if the given region is bounded or unbounded. - **Option 1:** The region is bounded. - **Option 2:** The region is unbounded. Additionally, find the coordinates of all corner points, if any exist. If an answer does not exist, please enter "DNE." **Instructions for Submission:** 1. **Smaller x-value coordinate:** - Format: \((x, y) = \) [Textbox for input] 2. **Larger x-value coordinate:** - Format: \((x, y) = \) [Textbox for input]
Transcribed Image Text:**Problem Statement:** Determine if the given region is bounded or unbounded. - **Option 1:** The region is bounded. - **Option 2:** The region is unbounded. Additionally, find the coordinates of all corner points, if any exist. If an answer does not exist, please enter "DNE." **Instructions for Submission:** 1. **Smaller x-value coordinate:** - Format: \((x, y) = \) [Textbox for input] 2. **Larger x-value coordinate:** - Format: \((x, y) = \) [Textbox for input]
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