Absolute value is a distance, not a sign remover

A tolerance check on a machined part, and the one reading of $|a|$ that makes every rule about it easy to see.

A workshop cuts steel pins that are meant to be 50 millimetres long. A pin passes inspection if it is within 0.2 millimetres of that, too long or too short makes no difference. Here is the inspection loop.

Look at what the test actually computes. It subtracts, throws away the sign, and compares. A pin at 49.8 and a pin at 50.2 get the same verdict, because they are the same distance from the target on opposite sides.

The definition

The absolute value of a real number $a$, written $|a|$, is the distance from $a$ to 0 on the number line. Distance is never negative, so $|a|$ is never negative. $|7| = 7$ and $|-7| = 7$, because both numbers sit seven steps from zero.

Written as a rule with two cases, that is:

That second line deserves a sentence of its own. If $a = -7$, then $-a$ means $-(-7)$, which is $7$. The minus sign in the definition is not making anything negative; it is undoing a negative. So $-a$ is not a negative quantity here, and reading it as one is the reason this definition looks wrong to people the first time they meet it.

The reading that makes everything else easy

Now the important step. The inspection loop did not compute $|length|$; it computed $|length - 50|$. In general, $|a - b|$ is the distance between $a$ and $b$ on the number line. That is the reading to carry for the rest of your mathematical life.

Distance does not care which end you start from, so $|a - b| = |b - a|$. Check it: $|3 - 10| = |-7| = 7$, and $|10 - 3| = |7| = 7$.

The picture is a V with its point at $c$. Being within $d$ of $c$ means being in the part of the V below the dashed line, and that part is the interval from $c - d$ to $c + d$. So the pin test $|L - 50| \le 0.2$ is exactly the interval $[49.8, 50.2]$, arrived at by reading, not by algebra.

The properties, and why each one holds

Five facts follow from the definition. None of them needs memorising if you keep hold of the distance picture.

There is a sixth fact that does more damage than any of those five when it is forgotten, and it is about square roots. The symbol $\sqrt{\phantom{x}}$ always returns the non-negative root, so $\sqrt{25} = 5$ and never $-5$. Feed it a square and see what comes back.

The middle column matches the last column every time, and matches $x$ only when $x$ is not negative. That is the rule:

A worked example, every step

Write "$x$ is more than 3 away from $-1$" using absolute value, and then as intervals.

1. The distance between $x$ and $-1$ is $|x - (-1)|$, which tidies to $|x + 1|$, because subtracting a negative adds.
2. "More than 3 away" makes the statement $|x + 1| > 3$.
3. Read it on the line: the point $-1$ is the centre, and you must be further than 3 steps from it.
4. Three steps to the right of $-1$ lands on 2; three steps to the left lands on $-4$.
5. Further than that means beyond 2, or before $-4$, and neither endpoint itself counts because the distance there is exactly 3, not more than 3.
6. So the answer is $(-\infty, -4) \cup (2, \infty)$. Check one value: $x = 10$ gives $|11| = 11$, which is more than 3, and 10 is indeed in the second piece.