Absolute Value and the Number Line

Grade 6 · Math · Study Guide

Understand how to use the number line and find absolute values of integers.

The number line is a straight line where every number has a place. Zero sits in the middle, positive numbers stretch to the right, and negative numbers stretch to the left. Numbers farther from zero are larger in absolute value, even if they're negative.

Absolute value is the distance a number is from zero on the number line — and distance is always positive (or zero). We write absolute value using vertical bars: |−7| = 7 and |7| = 7. Both numbers are exactly 7 units from zero, so they have the same absolute value.

Ordering integers on a number line means placing them correctly from least to greatest. The further left a number is, the smaller it is. So −10 < −3 < 0 < 4 < 9. This might feel backward at first — it seems like −10 should be 'more' because the number itself is larger — but on the number line, left always means less.

Absolute value has real meaning. If you owe $15, your balance is −15. The absolute value, $15, tells you the size of the debt without the direction. In science, distance between two points on a number line is found by taking the absolute value of their difference.

Key terms

Absolute Value

The distance of a number from zero on the number line — always non-negative

Opposite Numbers

Two numbers the same distance from zero but on opposite sides, like −6 and 6

Inequality

A statement that compares two numbers using < (less than) or > (greater than)

Number Line

A line where every real number corresponds to a unique point

Flashcards

Absolute value

Distance from zero — always positive or zero. |−8| = 8

|0| = ?

0 — zero is 0 units from itself

Opposite numbers

Same absolute value, opposite signs — e.g. 5 and −5

Ordering negatives

More negative = further left = smaller value. −10 < −3

Distance between two points

Take the absolute value of their difference — |a − b|

Why absolute value is never negative

It represents distance, which has no direction

Practice questions

1. What is |−14|?

   1. 14
   2. −14
   3. 0
   4. 1/14
   Show answer

   Answer: 14

   Absolute value strips the sign. |−14| = 14 because −14 is 14 units from zero.

2. −8 > −3 because 8 is bigger than 3.

   Show answer

   Answer: False

   On the number line −8 is further left than −3, making it smaller. −8 < −3, not greater.

3. |−25| = ___

   Show answer

   Answer: 25

   |−25| = 25. The distance from −25 to zero is 25 units.

4. Which list is in order from least to greatest?

   1. −10, −4, 0, 6
   2. 0, −4, −10, 6
   3. 6, 0, −4, −10
   4. −4, −10, 0, 6
   Show answer

   Answer: −10, −4, 0, 6

   Least to greatest goes left to right on the number line: −10 is leftmost, then −4, then 0, then 6.

5. Two numbers have an absolute value of 9. What are they?

   1. 9 and −9
   2. 0 and 9
   3. 9 and 1/9
   4. −9 and −9
   Show answer

   Answer: 9 and −9

   Absolute value measures distance from zero. Both 9 and −9 are exactly 9 units from zero.

Fun facts

• The symbol for absolute value (| |) was introduced by mathematician Karl Weierstrass in the 1840s.
• Absolute value is used in GPS navigation to calculate the shortest distance between two locations, regardless of direction.

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