Modulus: distance from zero

Three quick cases to fix the idea:

• $|5| = 5$, because $5$ is five steps to the right of zero.
• $|-5| = 5$, because $-5$ is also five steps from zero — the direction does not matter.
• $|0| = 0$, because zero is where you started.

So the modulus throws away the minus sign and keeps the size.

That single sentence solves most modulus questions.

Cambridge asks for three skills in this topic:

1. solve equations such as $|2x - 5| = 7$;
2. sketch $y = |ax + b|$;
3. use $|a| = |b| \iff a^2 = b^2$ when both sides have a modulus.

The modulus asks a distance question: "how far is $ax + b$ from zero?"

A distance of $c$ happens in two places — $c$ to the right and $c$ to the left.

$ax + b = c \qquad \text{or} \qquad ax + b = -c$

So the method is:

1. Check that $c \ge 0$. If $c < 0$ there is no solution, because a distance cannot be negative.
2. Write the two equations above.
3. Solve each one separately with ordinary algebra.
4. Check both answers by substituting back into the original equation.

The forgotten step is the second equation. Dropping the modulus without writing the $-c$ case loses one of the two answers.