Faceted Curriculum Project/B1/Math Facet

Mathematical Introduction
Equalities and inequalities, involving only linear expressions together with the absolute value operation, and only one variable, describe subsets of $$\mathbb{R}$$. Examples of such equalities and inequalities, in order of increasing complexity, are the following:
 * 1) $$\vert x \vert = 2 $$
 * 2) $$\vert x + 1 \vert \leq 1 $$
 * 3) $$\vert x + 2 \vert > \vert x \vert $$
 * 4) $$\vert x + \vert 2 - x \vert \vert \leq 7 $$

Every such equation or inequality $$ \Phi[x] $$ describes a set of real numbers:

$$S(\Phi) = \{ x \in \mathbb{R} \mbox{ such that } \Phi[x] \} $$

The task approached in this block is: Given such a formula $$ \Phi[x] $$, describe the set $$ S(\Phi) $$. There are two sufficient descriptions which are commonly used:
 * 1) A graphical description, on a number line.
 * 2) A formulaic description, using an equivalent algebraic formula involving only boolean combinations of simple equalities and inequalities.

When asked to "solve" an equality or inequality such as $$ \Phi[x] $$, one is being asked to provide such a description.

Algorithmic Solubility
Every such equation or inequality $$ \Phi[x] $$ is equivalent to a boolean combination of simple equalities or inequalities. By a simple equality or inequality (in the variable $$ x $$), we mean an expression of the form $$ x \mathcal{R} c $$, where $$ c $$ is a real constant, and $$ \mathcal{R} $$ is a relation (equality or inequality).

The existence, in general, of such an equivalence can be proven inductively. Algorithmically, every formula $$ \Phi[x] $$, as described before, is equivalent to a formula of the type:

$$ \vert F(x) \vert \mathcal{R} G(x) $$,

where $$ F(x) $$ and $$ G(x) $$ are formulae involving linear expressions and absolute values as before, and where $$ \mathcal{R} $$ is a relation (equality or inequality). Such a formula is equivalent to the boolean combination of formulae:

$$ ((F(x) \mathcal{R} G(x)) \wedge (F(x) \geq 0) ) \vee ((-F(x) \mathcal{R} G(x)) \wedge (F(x) \leq 0)).$$

Moreover, the above boolean combination of formulae contains fewer applications of the absolute value operation than the original formula $$ \Phi[x] $$ (unless $$ \Phi[x] $$ contained no absolute value operations to begin with). It follows by induction that every formula $$ \Phi[x] $$ is equivalent to a boolean combination of formulae involving linear equations and inequalities and no absolute value operations. Such a boolean combination of formulae describes a collection of (open or closed, and possibly unbounded) intervals in $$ \mathbb{R} $$.