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.