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We are tasked with solving the inequality \( |y - 3| \leq 7 \) and then graphing the solution on the number line.

### Step-by-Step Solution:

1. **Interpret the absolute value inequality**:
   \[
   |y - 3| \leq 7
   \]
   This inequality means that the distance between \( y \) and 3 is at most 7. We can rewrite this as a compound inequality:
   \[
   -7 \leq y - 3 \leq 7
   \]

2. **Solve the inequality**:
   To isolate \( y \), we add 3 to all parts of the inequality:
   \[
   -7 + 3 \leq y \leq 7 + 3
   \]
   Simplifying:
   \[
   -4 \leq y \leq 10
   \]

3. **Interpret the result**:
   This tells us that \( y \) can take any value between \(-4\) and \(10\), inclusive.

### Graphing the solution on the number line:
- On the number line provided, we would place closed dots (because the inequality includes equality) at \(-4\) and \(10\), and shade the region in between.

This represents all the possible values of \( y \) that satisfy the inequality.

Let me know if you'd like to see the graph or if you have any questions!

---

Here are some related questions to consider:
1. How do you solve an absolute value inequality if the sign is \( \geq \) instead of \( \leq \)?
2. How would the solution change if the inequality were \( |y - 3| < 7 \) instead of \( \leq \)?
3. Can you graph inequalities on a number line if the variables are not isolated?
4. What is the difference between "strict" inequalities and "non-strict" inequalities when graphing?
5. How do you graph solutions that involve absolute value inequalities where the result is an "or" condition (e.g., \( |x| \geq 5 \))?

**Tip:** When graphing absolute value inequalities, always remember that the inequality represents a distance on the number line, which can be translated into a range of possible values.

Graph the solution to the inequality |y - 3| ≤ 7 on the number line.

Learn how to solve and graph the absolute value inequality |y - 3| ≤ 7 on a number line. This step-by-step solution explains how to rewrite the inequality as a compound inequality and graph the result. Suitable for students in Grades 8-10.

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