College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.7: Graphs Of Inequalities
Problem 8SC
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Question
![### Question 32: Solving Systems of Inequalities
**Select the correct answer.**
**Which graph shows the solution region of this system of inequalities?**
\[
\begin{cases}
y \geq 2\left(1.3\right)^x \\
y \leq -x^2 + 4x + 5
\end{cases}
\]
#### Options:
- **A.**
Description: This graph features two shaded regions. The first is bordered by the exponential curve \( y = 2(1.3)^x \), shaded in blue to the right of the line, including the curve itself since the inequality is \(\geq\). The second shaded region is bordered by the quadratic function \( y = -x^2 +4x + 5 \), shaded in red to the left of the parabola, indicating the region where \( y \leq -x^2 +4x + 5 \). The solution region appears where these two shaded areas overlap, shown in a darker color.
- **B.**
Description: This graph has two shaded regions as well. The area below the parabola \( y \leq -x^2 + 4x + 5 \) is shaded in red, while the region above the exponential curve \( y \geq 2(1.3)^x \) is shaded in blue. The overlapping area is incorrect, as it does not align with the system's solution.
- **C.**
Description: In this option, the region above the exponential curve \( y \geq 2(1.3)^x \) is shaded in blue. The region below the quadratic curve \( y \leq -x^2 + 4x + 5 \) is shaded in red. The intersection of these two shaded regions correctly highlights the solution area of the system of inequalities.
#### Correct Answer:
**A.**
**Explanation:**
The correct graph accurately represents the intersection of the regions defined by the system of inequalities. The solution region is where both conditions \( y \geq 2(1.3)^x \) and \( y \leq -x^2 + 4x + 5 \) are simultaneously satisfied.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc6f3c32-694d-43a3-9d98-4837519070f4%2F8082a02c-a2af-4f37-b948-c3887347cfe3%2Fwxvcgb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 32: Solving Systems of Inequalities
**Select the correct answer.**
**Which graph shows the solution region of this system of inequalities?**
\[
\begin{cases}
y \geq 2\left(1.3\right)^x \\
y \leq -x^2 + 4x + 5
\end{cases}
\]
#### Options:
- **A.**
Description: This graph features two shaded regions. The first is bordered by the exponential curve \( y = 2(1.3)^x \), shaded in blue to the right of the line, including the curve itself since the inequality is \(\geq\). The second shaded region is bordered by the quadratic function \( y = -x^2 +4x + 5 \), shaded in red to the left of the parabola, indicating the region where \( y \leq -x^2 +4x + 5 \). The solution region appears where these two shaded areas overlap, shown in a darker color.
- **B.**
Description: This graph has two shaded regions as well. The area below the parabola \( y \leq -x^2 + 4x + 5 \) is shaded in red, while the region above the exponential curve \( y \geq 2(1.3)^x \) is shaded in blue. The overlapping area is incorrect, as it does not align with the system's solution.
- **C.**
Description: In this option, the region above the exponential curve \( y \geq 2(1.3)^x \) is shaded in blue. The region below the quadratic curve \( y \leq -x^2 + 4x + 5 \) is shaded in red. The intersection of these two shaded regions correctly highlights the solution area of the system of inequalities.
#### Correct Answer:
**A.**
**Explanation:**
The correct graph accurately represents the intersection of the regions defined by the system of inequalities. The solution region is where both conditions \( y \geq 2(1.3)^x \) and \( y \leq -x^2 + 4x + 5 \) are simultaneously satisfied.
![**Educational Website Transcription**
---
**Multiple Choice Question:**
Review the following graphs and select the correct one based on the given conditions.
**Graph Options:**
**A.**
*Description:* Depicts intersecting curves with highlighted regions.
**B.**
*Description:* The graph shows a blue curve and a red curve intersecting at multiple points. The regions are shaded as follows: blue for the areas above the blue curve but below the red curve, red for areas above the red curve but below the blue curve, and purple for the overlapping sections. Specifically, there is a distinct white region between x=0 and x=2 where the curves intersect.
**C.**
*Description:* Similar to graph B, this graph shows blue and red curves intersecting. The shading pattern differs slightly, having a sharper intersection within the white region. The purple area is above and to the left, the blue area is more towards the bottom and left, and the red area is towards the right.
**D.**
*Description:* The graph features intersecting curves with shaded regions: blue, red, and an intersecting white region prominently visible in the middle. The areas are shaded similarly where the blue area is below, red is above, and the overlapping purple region is centered more towards the middle-right.
**Choice:**
- **A**
- **B**
- **C**
- **D**
**Navigation:**
[Reset] [Next]
**Explanation of Graphs:**
**Graph B:**
- Displays two intersecting curves, with one primarily rising and the other curving from negative to positive y-values.
- Specific shaded regions indicate different inequalities. The white region depicts an area bounded by both curves' intersections.
**Graph C:**
- Shows a sharper intersection with a distinct mix of shaded regions.
- The overlap mainly occurs in the middle region, creating a compact white area where the curves meet.
**Graph D:**
- Features large shaded areas with a significant white intersecting region.
- The curves intersect prominently in the middle section of the graph, dividing the area into distinctly colored regions.
---
*Click "Next" to proceed after making your selection or "Reset" to start over.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc6f3c32-694d-43a3-9d98-4837519070f4%2F8082a02c-a2af-4f37-b948-c3887347cfe3%2Fvqml75_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Transcription**
---
**Multiple Choice Question:**
Review the following graphs and select the correct one based on the given conditions.
**Graph Options:**
**A.**
*Description:* Depicts intersecting curves with highlighted regions.
**B.**
*Description:* The graph shows a blue curve and a red curve intersecting at multiple points. The regions are shaded as follows: blue for the areas above the blue curve but below the red curve, red for areas above the red curve but below the blue curve, and purple for the overlapping sections. Specifically, there is a distinct white region between x=0 and x=2 where the curves intersect.
**C.**
*Description:* Similar to graph B, this graph shows blue and red curves intersecting. The shading pattern differs slightly, having a sharper intersection within the white region. The purple area is above and to the left, the blue area is more towards the bottom and left, and the red area is towards the right.
**D.**
*Description:* The graph features intersecting curves with shaded regions: blue, red, and an intersecting white region prominently visible in the middle. The areas are shaded similarly where the blue area is below, red is above, and the overlapping purple region is centered more towards the middle-right.
**Choice:**
- **A**
- **B**
- **C**
- **D**
**Navigation:**
[Reset] [Next]
**Explanation of Graphs:**
**Graph B:**
- Displays two intersecting curves, with one primarily rising and the other curving from negative to positive y-values.
- Specific shaded regions indicate different inequalities. The white region depicts an area bounded by both curves' intersections.
**Graph C:**
- Shows a sharper intersection with a distinct mix of shaded regions.
- The overlap mainly occurs in the middle region, creating a compact white area where the curves meet.
**Graph D:**
- Features large shaded areas with a significant white intersecting region.
- The curves intersect prominently in the middle section of the graph, dividing the area into distinctly colored regions.
---
*Click "Next" to proceed after making your selection or "Reset" to start over.*
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