EQN-01-09

Absolute Value & the Number Line

Read absolute value as distance from zero, and use it to handle "either sign" situations.

~4 min read•Difficulty 2/5

∣ x ∣ = the distance of x from 0, always \geq 0

∣ - 7∣ = 7 ∣7∣ = 7 ∣0∣ = 0

Absolute value strips the sign because distance is never negative.

∣ x ∣ = 5 \to x = 5 OR x = - 5

An absolute-value equation usually has two answers. Always check for the negative one.

distance on the number line = ∣ a - b ∣

The order does not matter $: ∣3 - 9∣ = ∣9 - 3∣ = 6.$

Treat the bars like parentheses: finish the inside first, then take the absolute value.

∣3 - 8∣ + 2 = ∣ - 5∣ + 2 = 5 + 2 = 7

Worked examples

Example 1 (difficulty 2/5)

Evaluate

: ∣4 - 10∣ + ∣ - 3∣

◎ Predict the answer first:

Example 2 (difficulty 2/5)

How many values of x satisfy

∣ x ∣ = 6

◎ Predict the answer first:

Giving only the positive solution of

∣ x ∣ = k

Why: The positive value is the obvious one.

Fix: Include the negative

: x = \pm k .

Thinking

∣ - 5∣

could be

- 5

Why: The minus sign inside is carried out.

Fix: Absolute value always returns a non-negative result.