Justify each step in the proofs of this theorem. Assume each variable represents a real number: a - (-b) = a + b

Let's go through the proof of the theorem step by step, justifying each one. The theorem is:

**Theorem:** \( a - (-b) = a + b \)

**Proof:**

1. **Given Expression:**  
   Start with the left-hand side of the equation:
   \[
   a - (-b)
   \]
   **Justification:** This is the expression we want to simplify and show that it equals \( a + b \).

2. **Definition of Subtraction:**  
   Subtraction can be defined as the addition of the additive inverse. Specifically:
   \[
   a - (-b) = a + (-(-b))
   \]
   **Justification:** By definition, subtracting a number is equivalent to adding its opposite. Here, subtracting \(-b\) is the same as adding the opposite of \(-b\), which is written as \(-(-b)\).

3. **Double Negative Property:**  
   The opposite (or additive inverse) of \(-b\) is \(b\). So,
   \[
   a + (-(-b)) = a + b
   \]
   **Justification:** The property of real numbers states that the negative of a negative number is the original positive number. Therefore, \(-(-b) = b\).

4. **Conclusion:**
   \[
   a - (-b) = a + b
   \]
   **Justification:** We have simplified the original expression \(a - (-b)\) to \(a + b\), which completes the proof.

Thus, every step in the proof is justified, showing that \( a - (-b) = a + b \) is indeed true for all real numbers \(a\) and \(b\).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you prove that \( -(a + b) = -a - b \)?
2. Can you justify why subtracting a positive number from another is the same as adding its negative?
3. How does the distributive property of multiplication over addition relate to this theorem?
4. What are other important properties of real numbers that are used frequently in proofs?
5. Why is the double negative property significant in simplifying expressions?

**Tip:** Always start by identifying the basic definitions and properties relevant to the equation or theorem you are working with; this often simplifies the proof process.

Understand and justify each step in the proof of the theorem a - (-b) = a + b with detailed explanations. Learn about subtraction, additive inverses, and the double negative property. Suitable for high school students and anyone interested in real number operations.

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