Sketch the region that corresponds to the given inequality. -x + 3y 2 6 y y 10 10- Solution Set Solution Set -10 -5 10 10 -5 10 -5 -5 10 10 y y 10 10 Solution Set -10 -5 10 10 10
Sketch the region that corresponds to the given inequality. -x + 3y 2 6 y y 10 10- Solution Set Solution Set -10 -5 10 10 -5 10 -5 -5 10 10 y y 10 10 Solution Set -10 -5 10 10 10
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Educational Website Content: Solving Inequalities
#### Graphing Inequalities
**Objective:** Sketch the region that corresponds to the given inequality.
**Given Inequality:**
\[
-x + 3y \geq 6
\]
**Explanation and Graphical Analysis:**
The task is to identify the solution set for the inequality \(-x + 3y \geq 6\).
1. **Graph Components:**
- **Axes:** Each graph includes an \(x\)-axis and a \(y\)-axis ranging from \(-10\) to \(10\).
- **Boundary Line:** The inequality is transformed into the equation \(-x + 3y = 6\), which is used as the boundary line. In these graphs, the boundary line is rendered as a solid line because the inequality is \(\geq\), meaning the line itself is included in the solution set.
2. **Solution Sets:**
- **First Graph (Top Left):**
- The shaded region (solution set) is above and to the right of the line, including the boundary. The shading indicates all points that satisfy the inequality.
- **Second Graph (Top Right):**
- The shaded area represents a different interpretation. Ensure the correct region is selected, as this choice provides a potential solution set that satisfies the inequality.
- **Third Graph (Bottom Left):**
- Again, the shaded region offers a possible solution set. Verifying the direction of shading (above the line) against the inequality is crucial for accuracy.
- **Fourth Graph (Bottom Right):**
- This graph shows another solution set interpretation. The region above the boundary line should be highlighted, illustrating inclusivity for all points meeting the inequality criteria.
**Conclusion:**
The correct graph for the inequality \(-x + 3y \geq 6\) will have shading above and to the right of the boundary line, accurately representing all solutions. Verify by substituting points from the shaded area into the original inequality to confirm they satisfy the condition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa75b3e4-cb9e-42f8-9b91-2e6340659e7d%2Fd8a3b913-bead-4a3f-8d78-a7708dc8d473%2F97mpcp8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Website Content: Solving Inequalities
#### Graphing Inequalities
**Objective:** Sketch the region that corresponds to the given inequality.
**Given Inequality:**
\[
-x + 3y \geq 6
\]
**Explanation and Graphical Analysis:**
The task is to identify the solution set for the inequality \(-x + 3y \geq 6\).
1. **Graph Components:**
- **Axes:** Each graph includes an \(x\)-axis and a \(y\)-axis ranging from \(-10\) to \(10\).
- **Boundary Line:** The inequality is transformed into the equation \(-x + 3y = 6\), which is used as the boundary line. In these graphs, the boundary line is rendered as a solid line because the inequality is \(\geq\), meaning the line itself is included in the solution set.
2. **Solution Sets:**
- **First Graph (Top Left):**
- The shaded region (solution set) is above and to the right of the line, including the boundary. The shading indicates all points that satisfy the inequality.
- **Second Graph (Top Right):**
- The shaded area represents a different interpretation. Ensure the correct region is selected, as this choice provides a potential solution set that satisfies the inequality.
- **Third Graph (Bottom Left):**
- Again, the shaded region offers a possible solution set. Verifying the direction of shading (above the line) against the inequality is crucial for accuracy.
- **Fourth Graph (Bottom Right):**
- This graph shows another solution set interpretation. The region above the boundary line should be highlighted, illustrating inclusivity for all points meeting the inequality criteria.
**Conclusion:**
The correct graph for the inequality \(-x + 3y \geq 6\) will have shading above and to the right of the boundary line, accurately representing all solutions. Verify by substituting points from the shaded area into the original inequality to confirm they satisfy the condition.
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