Björn took a poll for an upcoming local election and found that 27%
 of voters were likely to vote for the incumbent. His poll has a 4%
 margin of error.

Which of the following equations represents the likely minimum and maximum percentages of voters, v
, that will vote for the incumbent?

Choose TWO correct answers
A.|27−4|=v

B.|4−v|=27

C.|4−27|=v

D.|v−27|=4

E.|v−4|=27

F.|27−v|=4

In this problem, the margin of error tells us that the actual percentage of voters likely to vote for the incumbent could be as much as 4% higher or lower than the reported 27%. We need to find two equations that represent this situation.

### Let's break it down:

1. **Maximum likely percentage**:  
   The highest percentage \( v \) could be is \( 27 + 4 = 31 \)%. 
   
2. **Minimum likely percentage**:  
   The lowest percentage \( v \) could be is \( 27 - 4 = 23 \)%.

The problem is asking for equations that express the relationship between \( v \) and the given numbers, taking into account the margin of error.

Now, let's check the options:

- **Option A**: \( |27 - 4| = v \)  
  This simplifies to \( |23| = v \), or \( v = 23 \), which matches the minimum percentage. **This is a correct option.**

- **Option B**: \( |4 - v| = 27 \)  
  This equation does not logically represent the situation, as it suggests that the difference between 4 and \( v \) is 27, which doesn't relate to the problem. **This is not correct.**

- **Option C**: \( |4 - 27| = v \)  
  This simplifies to \( |−23| = v \), or \( v = 23 \), which matches the minimum percentage, but it is redundant with Option A and does not cover the full range. **This is not correct.**

- **Option D**: \( v - 27 = 4 \)  
  This represents the maximum percentage, since \( v = 27 + 4 = 31 \). **This is a correct option.**

- **Option E**: \( v - 4 = 27 \)  
  This would imply \( v = 31 \), but this is not quite the right setup for the margin of error situation. **This is not correct.**

- **Option F**: \( 27 - v = 4 \)  
  This represents the minimum percentage, since \( v = 27 - 4 = 23 \). **This is a correct option.**

### Correct answers:
- **A**: \( |27 - 4| = v \)
- **D**: \( v - 27 = 4 \)
- **F**: \( 27 - v = 4 \)

Do you have any questions or want more details?

### Related questions:
1. How is the margin of error typically calculated in polls?
2. What is the meaning of the absolute value notation \( |x| \) in real-life contexts?
3. What would the result be if the margin of error increased to 5%?
4. How do we handle rounding when dealing with percentages and polling?
5. What other statistical factors affect the accuracy of poll results?

**Tip**: When working with percentages, always keep an eye on possible rounding errors, especially in large datasets.

Björn took a poll for an upcoming local election and found that 27% of voters were likely to vote for the incumbent. His poll has a 4% margin of error. Which of the following equations represents the likely minimum and maximum percentages of voters, v, that will vote for the incumbent? Choose TWO correct answers: A.|27−4|=v B.|4−v|=27 C.|4−27|=v D.|v−27|=4 E.|v−4|=27 F.|27−v|=4

Learn how to determine the minimum and maximum likely percentages of voters who will support a candidate using a poll's margin of error. This problem covers key algebraic concepts, including solving absolute value and linear equations, and is suitable for students in Grades 8-10.

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