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Let's go through the problems one by one and solve them:

### Exercises 1-4: Simplify the expression.

1. \(|-2|\)
   - The absolute value of \(-2\) is \(2\).
   - **Answer**: \(2\)

2. \(|-7| - |7|\)
   - \(|-7| = 7\) and \(|7| = 7\), so \(7 - 7 = 0\).
   - **Answer**: \(0\)

3. \(|-3 + 2|\)
   - First, simplify inside the absolute value: \(-3 + 2 = -1\).
   - \(|-1| = 1\).
   - **Answer**: \(1\)

4. \(-\frac{-15}{5}\)
   - Simplify the fraction first: \(\frac{-15}{5} = -3\).
   - The negative sign outside makes it \(-(-3) = 3\).
   - **Answer**: \(3\)

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### Exercises 5-8: Solve the equation and graph the solutions if possible.

5. \(|r| = 5\)
   - This gives two solutions: \(r = 5\) or \(r = -5\).
   - **Graph**: Mark points at \(r = 5\) and \(r = -5\) on the number line.

6. \(|q| = -7\)
   - Since absolute values cannot be negative, there is **no solution**.

7. \(|b - 2| = 5\)
   - This gives two cases:
     1. \(b - 2 = 5\), so \(b = 7\).
     2. \(b - 2 = -5\), so \(b = -3\).
   - **Graph**: Mark points at \(b = 7\) and \(b = -3\) on the number line.

8. \(|k + 6| = 9\)
   - This gives two cases:
     1. \(k + 6 = 9\), so \(k = 3\).
     2. \(k + 6 = -9\), so \(k = -15\).
   - **Graph**: Mark points at \(k = 3\) and \(k = -15\) on the number line.

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### Exercises 9-14: Solve the equation and check the solutions.

9. \(|-5p| = 35\)
   - Solve for \(p\): \(|-5p| = 35\) gives two cases:
     1. \(-5p = 35\), so \(p = -7\).
     2. \(-5p = -35\), so \(p = 7\).
   - **Answer**: \(p = -7\) or \(p = 7\).

10. \(\left| \frac{a}{3} \right| = 4\)
    - Solve for \(a\): \(\left| \frac{a}{3} \right| = 4\) gives two cases:
      1. \(\frac{a}{3} = 4\), so \(a = 12\).
      2. \(\frac{a}{3} = -4\), so \(a = -12\).
    - **Answer**: \(a = 12\) or \(a = -12\).

11. \(|8y - 3| = 13\)
    - Solve for \(y\): \(|8y - 3| = 13\) gives two cases:
      1. \(8y - 3 = 13\), so \(8y = 16\), and \(y = 2\).
      2. \(8y - 3 = -13\), so \(8y = -10\), and \(y = -\frac{5}{4}\).
    - **Answer**: \(y = 2\) or \(y = -\frac{5}{4}\).

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Would you like more details on any specific problem or topic?

Here are some related questions:
1. How do you determine whether an absolute value equation has no solution?
2. What happens when you take the absolute value of a positive or negative number?
3. How do you solve compound absolute value inequalities?
4. What are the steps to graphing absolute value equations?
5. How do you handle absolute values in equations involving variables?

**Tip**: Always isolate the absolute value term first before considering the two possible cases when solving absolute value equations.

Math problems involving absolute value equations and expressions.

Learn how to solve absolute value equations and expressions step by step, with examples including |x| = a, solving for variables, and graphing solutions on a number line. Perfect for middle school students (Grades 6-8).

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