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The document you uploaded contains algebraic exercises focused on properties of real numbers, absolute value, and equations. Here’s a brief summary of the content:

1. **Equations Involving Absolute Value**:
   - Solve various absolute value equations, such as:
     - \( |x| = 3 \)
     - \( |x - 3| = \sqrt{3} - 1 \)
     - \( |3 - x| = 3x - 1 \), etc.

2. **Proofs in Algebra**:
   - Demonstrate algebraic identities, such as:
     - \( -(a + b) = (-a) + (-b) \)
     - \( (ab)^{-1} = a^{-1}b^{-1} \)
     - \( a - (b + c) = a - b - c \), etc.

3. **Basic Properties of Absolute Value**:
   - Prove properties like:
     - \( |x| \geq 0 \)
     - \( |x - y| = |y - x| \)
     - \( |xy| = |x||y| \), etc.

4. **Example Proofs**:
   - Provide examples such as:
     - \( ||x| - |y|| \leq |x - y| \)
     - \( ||x| - |y|| \leq |x + y| \)

5. **Inequalities Involving Absolute Value**:
   - Address inequalities like:
     - \( |x| < a \implies -a < x < a \)
     - \( |x| > a \implies x > a \text{ or } x < -a \)

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Here are 5 related questions you might explore:

1. How do you solve absolute value inequalities involving multiple variables?
2. What is the geometric interpretation of \( |x - y| \leq |x| + |y| \)?
3. How do you prove the property \( |x + y| \leq |x| + |y| \) for real numbers?
4. What are the steps to solve \( |x| > a \) for different cases of \( a \)?
5. How can you use the definition of absolute value to prove \( |xy| = |x||y| \)?

**Tip**: When solving absolute value equations, always remember to consider both positive and negative cases.

The document you uploaded contains algebraic exercises focused on properties of real numbers, absolute value, and equations.

Explore algebraic exercises focused on properties of real numbers, absolute values, and solving equations. This content includes key formulas, proofs, and example problems involving absolute value equations and inequalities, suitable for high school students in Grades 9-12.

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