The real numbers come with an order, and that order behaves well under addition but needs care under multiplication: multiplying an inequality by a negative number turns it around. Absolute value measures distance from zero, which turns questions about nearness into inequalities.

Reading a thermometer, minus ten is colder than minus three even though ten is the larger number. The order on the negatives runs against the size, which is exactly what multiplying by a negative does to an inequality.

Absolute value forgets the side and keeps the distance, so $\lvert-3\rvert$ and $\lvert3\rvert$ are both 3. That is what lets $\lvert x-a\rvert<\varepsilon$ be read as "$x$ is within $\varepsilon$ of $a$", which is how nearness is written for the rest of the course.

The absolute value of a number is its distance from zero, so the distance between two numbers is the absolute value of their difference. Almost every statement in analysis is eventually an inequality between such distances.

• Adding the same number to both sides preserves an inequality.
• Multiplying both sides by a positive number preserves it; by a negative number it reverses.
• $\lvert x - y \rvert$ is the distance between $x$ and $y$.
• $\lvert x \rvert \le c$ says exactly .

Each number lies between the negative and the positive of its own absolute value. Adding those two statements traps the sum in the same way, and being trapped is what an absolute value bound means.

A number never exceeds its absolute value, nor falls below its negative.

The same holds for the second number.

Adding two inequalities of the same direction keeps that direction.

Being trapped between $-c$ and $c$ is precisely what an absolute value at most $c$ means.

Multiplying by minus one flipped the sign, and the larger number became the smaller.

$\lvert x - 3 \rvert < 1$ says $x$ lies within $1$ of $3$.

That is the interval from $2$ to $4$, not counting the ends.

The single condition becomes a pair, which is usually far easier to work with.

Equality holds when the two point the same way; otherwise the left side is strictly smaller.

The triangle inequality is usually used with a term inserted and subtracted, so that one distance becomes two.

• Adding to both sides preserves an inequality; multiplying by a negative reverses it.
• $\lvert x-y\rvert$ is the distance from x to y, and $\lvert x\rvert\le c$ is $-c\le x\le c$.

Great work! You now know how to:

• Manipulate an inequality, reversing it when multiplying by a negative
• Read an absolute value as a distance and unpack it into two inequalities
• Use the triangle inequality by inserting a middle point