Absolute Value

Definition: The absolute value of a number is its distance from zero on the number line.

Key Property: Because it represents a distance, absolute value can never be negative.

Examples:

• For a positive number, the absolute value is the number itself: $|5| = 5$

• For zero, the absolute value is $0$: $|0| = 0$

• For a negative number, the absolute value is the positive version: $|-3| = 3$

In general:

$$ |x| \geq 0 $$

To simplify an absolute value expression, follow these 2 steps:

1. Work out the calculation inside the bars first.

2. Then apply the absolute value to make the result non-negative.

Example 1:

$$
\begin{aligned}
|-2 \times 6 + 3| &= |-12 + 3| \\
&= |-9| = 9
\end{aligned}
$$

Example 2:

$$
\begin{aligned}
|-2 \times 6| + |3| &= |-12| + |3| \\
&= 12 + 3 = 15
\end{aligned}
$$

Notice the difference:

• In the first example, the whole expression is inside one set of bars.
• In the second example, there are two separate bars, so we take the absolute value of each part before adding.

Method: When adding two negative numbers, add their absolute values, then keep the negative sign.

Example 1: – 30 – 120 = ?

• • $30 + 120 = 150$

  • Keep the negative sign → $-150$

Example 2: – 25 – 60 = ?

• • $25 + 60 = 85$

  • Keep the negative sign → $-85$

Exam Tip: Adding two negatives always makes the result more negative.

Method: When the numbers have different signs, subtract the smaller absolute value from the larger one, then keep the sign of the number with the larger absolute value.

Example 1: – 65 + 15 = ?

• • $65 – 15 = 50$
  • Larger absolute value is $65$ (negative) → $-50$

Example 2: 25 – 130 = ?

• • $130 – 25 = 105$
  • Larger absolute value is $130$ (negative) → $-105$

Exam Tip: Different signs mean you subtract the absolute values and keep the sign of the bigger number.