https://file.aiquickdraw.com/mathbot/file/1726804362816m68r1amn.pdf

Let's break down and solve each problem step by step.

---

### Problem 1: Solve for \( n \)
\[
3 - 7 + 2|n - 4| - |n| = -9
\]
To solve this, follow these steps:

1. **Simplify the equation**:
   Combine the constants first: 
   \[
   3 - 7 = -4, \quad \text{so the equation becomes:} \quad -4 + 2|n - 4| - |n| = -9.
   \]

2. **Isolate the absolute values**:
   Move -4 to the right side:
   \[
   2|n - 4| - |n| = -9 + 4 = -5.
   \]
   Now solve for \( |n| \) and \( |n - 4| \) by breaking the problem into cases depending on the value of \( n \).

---

### Problem 3: Find the distance between \(-22\) and \(-5\)
1. **Absolute value expression**: 
   The distance between two numbers \(a\) and \(b\) is given by \( |a - b| \). So:
   \[
   \text{absolute value expression} = |-22 - (-5)| = |-22 + 5| = |-17|.
   \]
   
2. **Distance**: 
   The distance is the absolute value of \(-17\), which is:
   \[
   17.
   \]

---

### Problem 4: Find the distance between \(-18\) and \(7\)
1. **Absolute value expression**:
   Using the same formula for distance:
   \[
   \text{absolute value expression} = |-18 - 7| = |-25|.
   \]
   
2. **Distance**:
   The distance is:
   \[
   25.
   \]

---

### Problem 5: Simplify the expression
\[
-\frac{1}{3}(9y - 3) + [(-8y) + (8y)]
\]
1. **Distribute** the \(-\frac{1}{3}\) across the terms inside the parentheses:
   \[
   -\frac{1}{3} \cdot 9y = -3y, \quad -\frac{1}{3} \cdot (-3) = 1.
   \]
   So the expression becomes:
   \[
   -3y + 1 + [(-8y) + (8y)].
   \]

2. **Simplify the terms inside the brackets**:
   \[
   (-8y) + (8y) = 0.
   \]
   Thus, the expression simplifies to:
   \[
   -3y + 1.
   \]

---

### Problem 6: Simplify the expression
\[
28 \div 4 \cdot 6 + \frac{32}{42} \cdot 11 + 5 - 22
\]
1. **First, simplify the operations step by step**:
   \[
   28 \div 4 = 7, \quad 7 \cdot 6 = 42.
   \]
   So the expression becomes:
   \[
   42 + \frac{32}{42} \cdot 11 + 5 - 22.
   \]

2. **Simplify the fractions**:
   \[
   \frac{32}{42} \cdot 11 = \frac{352}{42}.
   \]
   Thus, the expression becomes:
   \[
   42 + \frac{352}{42} + 5 - 22.
   \]
   Combine the terms for the final result.

---

Do you have any questions or want more details for any specific problem?

Here are 5 follow-up questions:
1. How can we handle cases in absolute value equations?
2. What steps do we take when simplifying complex expressions?
3. How do we convert word problems into algebraic expressions?
4. Can you explain how to simplify exponents with negative bases?
5. What are common mistakes when simplifying fractional expressions?

**Tip**: Always simplify expressions step by step to avoid missing small details that can lead to incorrect answers.

Solve for n: 3 − 7 + 2|n − 4| − |n| = −9

Learn how to solve absolute value equations like 3 − 7 + 2|n − 4| − |n| = −9 with step-by-step solutions. This problem involves core algebra concepts such as absolute values and distance calculations. Suitable for students in Grades 9-11.

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