Foundations — Compound inequalities — AND, OR

This page assumes you know nothing. We build every symbol the parent topic note uses, in an order where each one leans only on the ones before it. Read top to bottom.

0. The number line — the stage everything happens on

Before any symbol, we need a picture to point at.

WHY the topic needs it: every inequality below describes a region — a stretch of this line. "AND" and "OR" are just recipes for combining two stretches. If you can't see the line, you can't see the overlap. See Number line representation.

1. The symbols $<$, $>$, $\le$, $\ge$ — comparing two numbers

WHY the "or equal" line matters: it decides whether the boundary point itself is included. That single dot being in or out is the difference between an open and a closed end, which we picture next.

2. $x$ and the idea of a solution set

WHY the picture uses two kinds of dot:

• An open circle (○) at $3$ means "$3$ is not included" — used for $<$ and $>$.
• A filled dot (●) means "included" — used for $\le$ and $\ge$.

The shaded ray is the whole solution set at a glance. This is exactly the tool the parent note calls "each inequality is its own ray."

3. Rays and intervals — naming a stretch of line

A single inequality shades a ray: a half-line starting at one point and running forever one way.

WHY the topic needs this: the parent note's answers are all written as intervals like $(1,5]$ or $(-\infty ,-2)\cup [3,\infty )$. The bracket shape carries the same information as the open/filled dot in the figure. Full details: Interval notation.

4. Sets, and the symbols $\cap$, $\cup$, $\varnothing$

An inequality's answer is a set — a bag of numbers. Two sets can be combined two ways, and these two ways are the entire secret of AND / OR.

5. The words AND, OR — the logic glue

WHY "inclusive" matters: in everyday speech "coffee or tea" sometimes means "not both." In maths, OR always allows both. That's why $x<5$ or $x>2$ is all real numbers — the middle numbers satisfy both, and OR happily accepts them.

6. The compact three-part form $a<x<b$

Once $<$ and AND are clear, this shorthand costs nothing:

7. $\mathbb{R}$ — the whole line

Prerequisite map

Equipment checklist

Cover the right side; can you answer each before revealing?

Connections

• Compound inequalities — AND, OR — the parent topic these foundations feed.
• Number line representation — the stage in §0.
• Linear inequalities in one variable — each single ray comes from one of these.
• Interval notation — the bracket shorthand of §3.
• Set operations — union and intersection — the $\cap$, $\cup$ backbone of §4.
• Absolute value inequalities — where AND/OR reappear from $|x|$.