Question
give me more questions about |x+11|=24 and the sum of the solutions to the given equation
Answers
GPT-4o mini
To explore the equation \( |x + 11| = 24 \), let's break it down into multiple questions that can further enhance understanding of the absolute value concept and related equations. Here are some questions you might find useful:
1. **Find the solutions to the equation:**
Solve \( |x + 11| = 24 \).
2. **Identify possible cases:**
What are the two cases we have when dealing with absolute value equations like \( |x + 11| = 24 \)?
3. **Write the equivalent equations:**
What are the two equations that arise from the absolute value equation \( |x + 11| = 24 \)?
4. **Calculate each solution:**
Solve the equations \( x + 11 = 24 \) and \( x + 11 = -24 \).
5. **Verify the solutions:**
Substitute your solutions back into the original equation \( |x + 11| = 24 \) to verify that they are correct.
6. **Calculate the sum of the solutions:**
What is the sum of the solutions found in the previous step?
7. **Explain the significance of the absolute value in this context:**
Why does the absolute value affect the number of solutions to the equation?
8. **Graph the equation:**
How would you represent the equation \( |x + 11| = 24 \) on a number line? What points would you plot?
9. **Analyze the transformations:**
Describe how the graph of \( |x| \) is transformed to obtain the graph of \( |x + 11| \).
10. **Consider different equations:**
How would the solutions to the equation \( |x + 11| = 10 \) compare to those of \( |x + 11| = 24 \)?
### Sum of the Solutions
For the equation \( |x + 11| = 24 \):
1. The two cases that arise are:
- \( x + 11 = 24 \)
- \( x + 11 = -24 \)
2. Solving these gives:
- From \( x + 11 = 24 \) → \( x = 24 - 11 = 13 \)
- From \( x + 11 = -24 \) → \( x = -24 - 11 = -35 \)
3. Therefore, the solutions are \( x = 13 \) and \( x = -35 \).
4. To find the sum of the solutions:
\[
13 + (-35) = 13 - 35 = -22.
\]
Thus, the sum of the solutions to the given equation \( |x + 11| = 24 \) is \(-22\).
1. **Find the solutions to the equation:**
Solve \( |x + 11| = 24 \).
2. **Identify possible cases:**
What are the two cases we have when dealing with absolute value equations like \( |x + 11| = 24 \)?
3. **Write the equivalent equations:**
What are the two equations that arise from the absolute value equation \( |x + 11| = 24 \)?
4. **Calculate each solution:**
Solve the equations \( x + 11 = 24 \) and \( x + 11 = -24 \).
5. **Verify the solutions:**
Substitute your solutions back into the original equation \( |x + 11| = 24 \) to verify that they are correct.
6. **Calculate the sum of the solutions:**
What is the sum of the solutions found in the previous step?
7. **Explain the significance of the absolute value in this context:**
Why does the absolute value affect the number of solutions to the equation?
8. **Graph the equation:**
How would you represent the equation \( |x + 11| = 24 \) on a number line? What points would you plot?
9. **Analyze the transformations:**
Describe how the graph of \( |x| \) is transformed to obtain the graph of \( |x + 11| \).
10. **Consider different equations:**
How would the solutions to the equation \( |x + 11| = 10 \) compare to those of \( |x + 11| = 24 \)?
### Sum of the Solutions
For the equation \( |x + 11| = 24 \):
1. The two cases that arise are:
- \( x + 11 = 24 \)
- \( x + 11 = -24 \)
2. Solving these gives:
- From \( x + 11 = 24 \) → \( x = 24 - 11 = 13 \)
- From \( x + 11 = -24 \) → \( x = -24 - 11 = -35 \)
3. Therefore, the solutions are \( x = 13 \) and \( x = -35 \).
4. To find the sum of the solutions:
\[
13 + (-35) = 13 - 35 = -22.
\]
Thus, the sum of the solutions to the given equation \( |x + 11| = 24 \) is \(-22\).