Absolute value

The absolute value of a number, written $|x|$, measures how far $x$ is from zero on the number line - regardless of direction. Whether we move $5$ units to the left or $5$ units to the right, the distance is the same: $5$. So $|{-5}| = 5$ and $|5| = 5$.

Because distance is never negative, $|x| \geq 0$ for every value of $x$.

The formal piecewise definition pins this down:

We can extend the distance idea beyond zero. The expression $|x - a|$ gives the distance between $x$ and $a$ on the number line. For example, $|2 - 7| = |{-5}| = 5$, matching the $5$ units between $2$ and $7$. The order does not matter: $|7 - 2| = |5| = 5$ gives the same result.

Evaluate $|{-7}|$ and $|3|$ using the piecewise definition. Then compute $|x - 4|$ when $x = -1$ and interpret the result as a distance.

Solution

Evaluating $|{-7}|$:

Since $-7 < 0$, we use the second piece of the definition: $|x| = -x$. Substituting:

Evaluating $|3|$:

Since $3 \geq 0$, we use the first piece: $|x| = x$. So:

Computing $|x - 4|$ when $x = -1$:

Substituting $x = -1$:

Since $-5 < 0$, we apply $|x| = -x$:

The result $5$ confirms the distance interpretation: $-1$ and $4$ are $5$ units apart on the number line.

Absolute value interacts neatly with multiplication and division through two rules.

The absolute value of a product equals the product of the absolute values. It does not matter whether we take the absolute value before or after multiplying.

The same idea works for division - take the absolute value of each part separately.

Simplify $|{-4} \cdot 7|$ and $\left|\dfrac{-12}{3}\right|$ using the product and quotient rules.

Solution

Product rule: $|{-4} \cdot 7|$

Applying $|ab| = |a| \cdot |b|$:

We can verify: $-4 \cdot 7 = -28$, and $|{-28}| = 28$.

Quotient rule: $\left|\dfrac{-12}{3}\right|$

Applying $\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}$:

Again we can verify: $\dfrac{-12}{3} = -4$, and $|{-4}| = 4$.

We saw that $|a + b|$ is not equal to $|a| + |b|$ in general. The triangle inequality tells us the precise relationship: the left side is always less than or equal to the right.

In other words, the absolute value of a sum is at most the sum of the absolute values. The right-hand side acts as an upper bound on $|a + b|$.

The diagram gives both sides of the inequality a geometric meaning: $|a| + |b|$ is the total distance covered by the two steps, while $|a + b|$ is the distance from $0$ to the final position.

When the steps point the same way, no ground is retraced and the two sides agree. When they oppose, the second step retraces part of the first, so the final position is closer to $0$ than the total distance covered.

In terms of signs: equality holds exactly when $a$ and $b$ have the same sign (or either is zero), and opposite signs force strict inequality.

To verify the triangle inequality for a pair of values, compute $\lvert a + b \rvert$ and $\lvert a \rvert + \lvert b \rvert$ separately, then check the inequality holds.

1. $a = 3$, $b = 5$ (same sign): $\lvert 3 + 5 \rvert = 8$ and $\lvert 3 \rvert + \lvert 5 \rvert = 8$. We get $8 \leq 8$ - equality.
2. $a = 3$, $b = -5$ (opposite signs): $\lvert 3 + (-5) \rvert = 2$ and $\lvert 3 \rvert + \lvert{-5}\rvert = 8$. We get $2 \leq 8$ - strict inequality, because the opposite signs cause partial cancellation.
3. $a = -4$, $b = 0$: $\lvert{-4} + 0\rvert = 4$ and $\lvert{-4}\rvert + \lvert 0 \rvert = 4$. We get $4 \leq 4$ - equality again.