Understanding the Context

First, we identify the essential lines:

• Line 1: $ x + y = 0 $, which simplifies to $ y = -x $ — a diagonal line passing through the origin at a 45° angle downward into the left and right quadrants.
  
• Line 2: $ x - y = 0 $, equivalent to $ y = x $ — a diagonal line ascending into the upper-right and lower-left quadrants.

These lines intersect at the origin $(0,0)$, but unlike the familiar $ x = 0 $ and $ y = 0 $ axes, they form a finer division not aligned with coordinate directions. Their slopes are $ m_1 = -1 $ and $ m_2 = 1 $, and they create four acute angular regions around the origin.

How the Lines Divide the Plane: The Four Quadrants

Key Insights

By definition, two intersecting lines split the plane into four regions, analogous to coordinate quadrants. Applying this to the lines $ x + y = 0 $ and $ x - y = 0 $, we define the quadrants based on the sign of expressions on both sides of each line.

Quadrant I: Above both lines $ y > x $ and $ y > -x $

Above $ y = x $ and above $ y = -x $: this region lies strictly in the upper-right and upper-left sectors, where $ y > x $ and $ y > -x $ coexist. This area satisfies both inequalities simultaneously.

Quadrant II: Below $ y = x $ and above $ y = -x $

Below $ y = x $ but above $ y = -x $. This region lies below the rising line but above the downward-sloping diagonal, capturing points where $ -x < y < x $, applicable especially in the second and fourth quadrants.

Quadrant III: Below $ y = x $ and below $ y = -x $

Below both lines — often considered the lower-left and lower-right zone — this quadrant satisfies $ y < x $ and $ y < -x $. Notably, in this region $ x > |y| $, and absolute values dominate region elections.

Quadrant IV: Above $ y = x $ and below $ y = -x $

Above $ y = x $ yet below $ y = -x $? Rare in standard classification, but mathematically, this intersection region only exists where $ x < y < -x $, which is geometrically empty unless limited to negative $ x $. It reflects a critical conceptual edge where quadrant definitions converge and conflict.

Final Thoughts

Important note: Unlike the standard axial quadrants, the regions formed by $ x + y = 0 $ and $ x - y = 0 $ exhibit symmetry under reflection across both axes and diagonals. The four quadrants are not merely positional—they underscore deeper algebraic symmetry tied to sign changes and parity.

Geometric Significance and Applications

Understanding this quadrant system enhances:

• Algebraic reasoning: Solving inequalities involving $ x + y $ and $ x - y $ becomes intuitive when visualizing sign regions.
  
• Graphing and plotting: Recognizing these zones helps in analyzing symmetric functions and piecewise-defined relationships.
  
• Computer graphics and geometry: These divisions inform ray tracing, collision detection, and spatial partitioning algorithms.
  
• Physics and navigation: Directional vector analysis benefits from decomposing motion in rotating frames or diagonal reference systems.

Practical Visualization Tips

To best understand the four quadrants: