Absolute Value Operations and Inequalities

Category: Coordinate Analysis & Inequities

1. Distance From Zero

The Absolute Value of a number, denoted as |x|, tracks its magnitude or physical distance from zero on a number line, completely disregarding its directional sign. Because distance cannot be negative, absolute values always yield positive scalars.

2. Splitting Absolute Equations

Because an absolute distance can extend in either a positive or negative direction along a coordinate line, solving an equation like |x| = c requires splitting the expression into two separate, independent algebraic branches: x = c and x = -c.

3. Proof of the Triangle Inequality Theorem

The Triangle Inequality is a foundational rule in real analysis and algebra. It states that the absolute value of a sum of two terms is always less than or equal to the sum of their individual absolute values: |a + b| ≤ |a| + |b|.

Theorem to prove: |a + b| ≤ |a| + |b|

Step 1: Square the absolute expression, utilizing the identity |x|² = x²:
|a + b|² = (a + b)² = a² + 2ab + b²

Step 2: Replace individual squared terms with their absolute equivalents:
|a + b|² = |a|² + 2ab + |b|²

Step 3: Establish a bounding inequality. A basic product 'ab' is always less than or equal to its absolute value magnitude '|ab|':
ab ≤ |ab| = |a|·|b|

Step 4: Substitute this boundary parameter back into our expanded equation:
|a + b|² ≤ |a|² + 2|a||b| + |b|²

Step 5: Compress the right side back into a perfect square binomial expansion:
|a + b|² ≤ (|a| + |b|)²

Step 6: Take the positive square root of both sides of the inequality:
|a + b| ≤ |a| + |b|

This proof demonstrates how directional sign dynamics establish the mathematical boundaries for spatial coordinate maps.

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