Follow the steps outlined here to prove the triangle...

Follow the steps outlined here to prove the triangle inequality: $|a+b| \leq|a|+|b|$ for any real numbers $a$ and $b$ : (a) Argue that for any real number $x,|x|=\sqrt{x^{2}}$. (b) Show that $(a+b)^{2} \leq(|a|+|b|)^{2}$. (Hint: Start on the left-hand side, multiply out the expression, and use the fact that $a \leq|a|$ and $b \leq|b| .)$ (c) Take the square root of both sides of the inequality from part (b) (this is valid since both sides are positive), and use part (a) to show that $|a+b| \leq|a|+|b|$.

Follow the steps outlined here to prove the triangle inequality: $|a+b| \leq|a|+|b|$ for any real numbers $a$ and $b$ :
(a) Argue that for any real number $x,|x|=\sqrt{x^{2}}$.
(b) Show that $(a+b)^{2} \leq(|a|+|b|)^{2}$. (Hint: Start on the left-hand side, multiply out the expression, and use the fact that $a \leq|a|$ and $b \leq|b| .)$
(c) Take the square root of both sides of the inequality from part (b) (this is valid since both sides are positive), and use part (a) to show that $|a+b| \leq|a|+|b|$.

Key Concepts

Triangle Inequality Principle

The triangle inequality, which states that the absolute value of the sum of two numbers is less than or equal to the sum of their absolute values, is a fundamental concept in analysis and metric spaces. It reflects the idea that the direct path between two points is always shorter than or equal to any indirect path, making it essential for understanding distances and norms in more abstract settings.

Absolute Value as a Square Root

The absolute value of a real number is defined as its distance from zero on the number line, which can be mathematically expressed as |x| = ?(x²). This formulation is crucial because it allows us to use properties of square roots and squares, ensuring that expressions involving |x| are always non-negative.

Properties of Squaring and Non-Negativity

Squaring any real number results in a non-negative value, a property that is exploited in many algebraic proofs including the triangle inequality. This is important because it provides a guarantee that when both sides of an equation or inequality are squared, they remain within the realm of non-negative expressions, permitting valid operations such as taking square roots.

Algebraic Expansion and Manipulation

Expanding and rearranging expressions is a key algebraic skill. In the context of proving the triangle inequality, expanding the square of a sum and then comparing terms using known inequalities (like a ? |a|) demonstrates how careful algebraic manipulation supports the derivation of broader inequality principles.

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