1. Let's state the problem: We want to understand why $$|x| + |y|$$ squared is different from $$|x + y|$$ squared. 2. The expressions are: - $$ (|x| + |y|)^2 $$ - $$ |x + y|^2 $$ 3. Recall the properties of absolute values: - $$|a| \geq 0$$ for any real number $$a$$. - $$|a|^2 = a^2$$. - The triangle inequality: $$|x + y| \leq |x| + |y|$$. 4. Let's expand $$ (|x| + |y|)^2 $$: $$ (|x| + |y|)^2 = |x|^2 + 2|x||y| + |y|^2 = x^2 + 2|x||y| + y^2 $$ 5. Now, $$ |x + y|^2 = (x + y)^2 = x^2 + 2xy + y^2 $$ 6. Comparing the two: - $$ (|x| + |y|)^2 = x^2 + 2|x||y| + y^2 $$ - $$ |x + y|^2 = x^2 + 2xy + y^2 $$ 7. The difference lies in the middle term: - $$2|x||y|$$ vs. $$2xy$$. 8. Since $$|x||y| \geq |xy|$$, and $$xy$$ can be negative if $$x$$ and $$y$$ have opposite signs, the two expressions are generally different. 9. For example, if $$x = -1$$ and $$y = 1$$: - $$ (|x| + |y|)^2 = (1 + 1)^2 = 4 $$ - $$ |x + y|^2 = |0|^2 = 0 $$ 10. This shows why $$ (|x| + |y|)^2 $$ is generally greater than or equal to $$ |x + y|^2 $$. Final answer: $$ (|x| + |y|)^2 \neq |x + y|^2 $$ in general because the middle terms differ due to the absolute value affecting the sign of the product.