### Problem StatementDarren is training for a triathlon. Each week, he goes to the gym for a maximum of 20 hours. He spends at least 8 of those hours weightlifting (x). Darren wants to spend no more than 15 hours doing cardio exercises (y). Write a system of linear inequalities that represents the restrictions of this scenario. Choose the graph that represents the solutions to the system of linear inequalities.### System of Linear Inequalities1. $ x + y \leq 20 $ (Total gym hours cannot exceed 20)2. $ x \geq 8 $ (At least 8 hours of weightlifting)3. $ y \leq 15 $ (No more than 15 hours of cardio)### Graph Descriptions1. **Top-Left Graph:** - Axes range from 0 to 20 on both x (weightlifting) and y (cardio). - A shaded region on the graph represents feasible solutions. - The shaded area forms a polygon bounded by $ x = 8 $, $ y = 15 $, and $ x + y = 20 $. - The overlapping area of purple indicates the solution region satisfying all inequalities.2. **Top-Right Graph:** - Similar axes ranging from 0 to 20. - The blue shaded region highlights the feasible solutions. - The overlap is less clear in this graph compared to others, with less emphasis on the feasible solution space.3. **Bottom-Left Graph:** - Axes display a 0 to 20 range for both variables. - The purple area marks the overlap that is the solution to the inequalities. - The overlap appears smaller, focusing on different constraints.4. **Bottom-Right Graph:** - With the same axes scale, this graph uses red to show feasible solutions. - The shading prominently reflects where the constraints overlap, indicating compliance with all inequalities. - Emphasizes meeting the requirements of spending at least 8 hours weightlifting and no more than 15 on cardio.### ConclusionEach graph represents different configurations of the given inequalities. The correct graph will clearly depict the overlapping area constrained by all three inequalities. The intersection of feasible weightlifting and cardio hours within the restrictions is key in identifying the right graph. **Problem Statement:**Lena is a pediatrician. In January, she saw a total of more than 160 patients, consisting of girls ($x$) and boys ($y$). In February, she saw fewer than 100 patients, consisting of girls ($x$) and boys ($y$).The inequalities graph below represents this situation. Which of the following statements describes the situation?**Graph Analysis:**The graph displays a coordinate system with $x$ and $y$ axes both labeled from 0 to 250. There are two shaded regions and a white region. The graph lines represent the borders for the inequalities:- The line on the left with negative slope and within the blue region.- The line at the top bordering the green-shaded region.**Statements for Consideration:**- The inequality $x + y > 100$ represents the total number of patients Lena saw in February.- The inequality $x + y < 160$ represents the total number of patients Lena saw in January.- The white area is the feasible region.- There is no solution to the system of inequalities.The blue shading indicates where $x + y > 160$ is true, the green shading where $x + y < 100$ is true, and the overlapping region is the white area, which is not feasible for both conditions.

Answered: Image /qna-images/answer/e95f98de-d6b6-447f-a884-4da69fc692d7.jpg

Darren is training for a triathlon. Each week, he goes to the gym for a maximum of 20 hours. He spends at least 8 of those hours weightlifting (z). He wants to spend no more than 15 hours doing cardio exercises (y). Write a system of linear inequalities that represents the restrictions of this scenario. Choose the graph that represents th solutions to the system of linear inequalities. All changes V 20 ' 20 15 15 10 10 5. 20 V 20 15- 10 20

Darren is training for a triathlon. Each week, he goes to the gym for a maximum of 20 hours. He spends at least 8 of those hours weightlifting (z). He wants to spend no more than 15 hours doing cardio exercises (y). Write a system of linear inequalities that represents the restrictions of this scenario. Choose the graph that represents th solutions to the system of linear inequalities. All changes V 20 ' 20 15 15 10 10 5. 20 V 20 15- 10 20

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Problem Statement

Darren is training for a triathlon. Each week, he goes to the gym for a maximum of 20 hours. He spends at least 8 of those hours weightlifting (x). Darren wants to spend no more than 15 hours doing cardio exercises (y). Write a system of linear inequalities that represents the restrictions of this scenario. Choose the graph that represents the solutions to the system of linear inequalities.

### System of Linear Inequalities

1. $ x + y \leq 20 $ (Total gym hours cannot exceed 20)
2. $ x \geq 8 $ (At least 8 hours of weightlifting)
3. $ y \leq 15 $ (No more than 15 hours of cardio)

### Graph Descriptions

1. **Top-Left Graph:**
- Axes range from 0 to 20 on both x (weightlifting) and y (cardio).
- A shaded region on the graph represents feasible solutions.
- The shaded area forms a polygon bounded by $ x = 8 $, $ y = 15 $, and $ x + y = 20 $.
- The overlapping area of purple indicates the solution region satisfying all inequalities.

2. **Top-Right Graph:**
- Similar axes ranging from 0 to 20.
- The blue shaded region highlights the feasible solutions.
- The overlap is less clear in this graph compared to others, with less emphasis on the feasible solution space.

3. **Bottom-Left Graph:**
- Axes display a 0 to 20 range for both variables.
- The purple area marks the overlap that is the solution to the inequalities.
- The overlap appears smaller, focusing on different constraints.

4. **Bottom-Right Graph:**
- With the same axes scale, this graph uses red to show feasible solutions.
- The shading prominently reflects where the constraints overlap, indicating compliance with all inequalities.
- Emphasizes meeting the requirements of spending at least 8 hours weightlifting and no more than 15 on cardio.

### Conclusion

Each graph represents different configurations of the given inequalities. The correct graph will clearly depict the overlapping area constrained by all three inequalities. The intersection of feasible weightlifting and cardio hours within the restrictions is key in identifying the right graph.

**Problem Statement:**

Lena is a pediatrician. In January, she saw a total of more than 160 patients, consisting of girls ($x$) and boys ($y$). In February, she saw fewer than 100 patients, consisting of girls ($x$) and boys ($y$).

The inequalities graph below represents this situation. Which of the following statements describes the situation?

**Graph Analysis:**

The graph displays a coordinate system with $x$ and $y$ axes both labeled from 0 to 250. There are two shaded regions and a white region. The graph lines represent the borders for the inequalities:
- The line on the left with negative slope and within the blue region.
- The line at the top bordering the green-shaded region.

**Statements for Consideration:**

- The inequality $x + y > 100$ represents the total number of patients Lena saw in February.
- The inequality $x + y < 160$ represents the total number of patients Lena saw in January.
- The white area is the feasible region.
- There is no solution to the system of inequalities.

The blue shading indicates where $x + y > 160$ is true, the green shading where $x + y < 100$ is true, and the overlapping region is the white area, which is not feasible for both conditions.

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