To prove that \( |a + b| \leq \sqrt{2} \) given \( a^2 + b^2 = 1 \), let's proceed with the following steps:

### Step 1: Express the sum \( |a + b| \) in terms of \( a \) and \( b \).
The absolute value of the sum is given by:
\[
|a + b| = \sqrt{(a + b)^2}
\]
Expanding the square:
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Using the given condition \( a^2 + b^2 = 1 \):
\[
(a + b)^2 = 1 + 2ab
\]
Thus, we have:
\[
|a + b| = \sqrt{1 + 2ab}
\]

### Step 2: Analyze the maximum possible value of \( |a + b| \).
The expression \( |a + b| = \sqrt{1 + 2ab} \) is maximized when \( ab \) is maximized. Since \( a^2 + b^2 = 1 \), by the Cauchy-Schwarz inequality (or using the fact that \( ab \) is maximized when \( a = b \)), the maximum value of \( ab \) is \( \frac{1}{2} \).

Substituting \( ab = \frac{1}{2} \) into the expression for \( |a + b| \):
\[
|a + b| = \sqrt{1 + 2 \times \frac{1}{2}} = \sqrt{1 + 1} = \sqrt{2}
\]

### Conclusion:
Since the expression \( \sqrt{1 + 2ab} \) achieves its maximum value when \( ab \) is at its maximum, the maximum value of \( |a + b| \) is \( \sqrt{2} \). Therefore:
\[
|a + b| \leq \sqrt{2}
\]

This completes the proof that \( |a + b| \leq \sqrt{2} \) given \( a^2 + b^2 = 1 \).

---

Do you want more details or have any questions? 

Here are 5 related questions that might interest you:
1. How does the Cauchy-Schwarz inequality relate to maximizing \( ab \)?
2. Can we generalize the result for \( a^2 + b^2 = r^2 \) instead of \( 1 \)?
3. What are the geometric interpretations of \( a^2 + b^2 = 1 \) and \( |a + b| \)?
4. How does this result apply to vectors in two-dimensional space?
5. What happens if \( a^2 + b^2 \neq 1 \)?

**Tip:** When dealing with inequalities involving absolute values, consider squaring both sides to simplify the expression, but remember to check for extraneous solutions!

Learn how to prove the inequality |a + b| ≤ √2 given that a² + b² = 1. This step-by-step solution explores key algebraic concepts, including the Cauchy-Schwarz inequality, and uses geometric insights to reach the result. Suitable for students in Grades 10-12.

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