1.D Inequalities - JulTob/Mathematics GitHub Wiki

Infinite Intervals

Let $a$ be any real number. The set of all real numbers $x$ such that:

• $x < a$,
• $x \leq a$,
• $x > a$, or
• $x \geq a$

is called an infinite interval. These intervals extend indefinitely in one direction and are written using the symbol for infinity $\infty$, though $\infty$ is not a number but a formal symbol representing unboundedness:

• $(-\infty, a)$
• $(-\infty, a]$
• $(a, +\infty)$
• $[a, +\infty)$

Such notation serves as a compact encoding of range conditions. Its utility lies not only in its formal precision but in its alignment with conceptual intuitions about limits and boundaries.

Constants and Variables

Consider the double inequality:

$a < x < b$

Here:

• $a$, $b$ are constants, i.e., fixed reference values.
• $x$ is a variable, which may assume values within a specified domain.

A variable does not range over all numbers indiscriminately. Its field of variation is contextual—defined explicitly by conditions or implicitly by convention.

Examples of Variable Fields:

• If $x$ denotes a volume in a 10-liter container: $0 \leq x \leq 10$
• If $x$ is a day in July: $x \in {1, 2, \dots, 31}$
• If $x$ is a label for books in a collection: $x \in {1, 2, \dots, 10}$

This framing invites interpretation of variables as placeholders in structured domains, akin to roles in syntactic phrases.

Transformations and Validity

Solving inequalities requires transformation rules that preserve truth values. Some of these operations preserve the direction of inequality; others invert it.

Operation	Direction Preserved?
Add/subtract same number on both sides	✅
Multiply/divide by positive number	✅
Multiply/divide by negative number	❌ (flip direction)

For instance:

$-2x > 6 \Rightarrow x < -3$

Here, dividing by $-2$ inverts the inequality. The rule is not merely mechanical; it reflects the reversal of order under scalar reflection.

Interpretation and Application

Inequalities define conditions, not identities. In contrast with equations, which represent balance, inequalities delineate regions of possibility. In applications:

• In optimization, they encode constraints.
• In analysis, they describe bounds.
• In probability, they delimit events.

Understanding inequalities requires a shift from solving to interpreting. Their form reflects comparison, and their function—like that of grammar in language—is to structure relational meaning.