Mirrored distances from zero on a glowing number line, illustrating absolute value.
Unit 3 · Negative Numbers & the Coordinate Plane · 6.NS

Lesson 3.2 — Absolute Value & Comparing

How far a number is from zero — and how to tell which of two numbers is greater.

🎙️ Narration script

Welcome back! Today we're going to look at two ideas that sound similar but are actually different: absolute value, and comparing numbers.

Let's start with absolute value. Absolute value is just a number's distance from zero. And since distance is always positive, absolute value is always positive, or zero. We write it with two little bars on each side. So the absolute value of negative seven is seven. And the absolute value of seven is also seven. Why the same? Because distance doesn't care which direction you traveled. Negative seven is seven steps from zero, and so is positive seven.

Think about a diver in the ocean. A diver at negative forty meters is forty meters away from the surface. That forty is the absolute value. It tells you the size, the magnitude, no matter the direction.

Now, comparing numbers is a different thing. Comparing is all about position on the number line, and farther to the right means greater.

Here's where people get tangled up, so listen closely. Is negative ten bigger than negative two? No! Negative ten is actually smaller, because it sits farther to the left. It's colder. But the absolute value of negative ten is ten, which is bigger than the absolute value of negative two. So a bigger absolute value does not mean a bigger number.

Let's say that clearly. Negative eight is less than negative three. But the absolute value of negative eight is greater than the absolute value of negative three.

Quick recap. Absolute value is distance from zero, and it's never negative. Comparing is about position, where farther right is greater. Just don't mix the two up. You've got this!

1 Core idea

Absolute value is a number's distance from zero — so it's always positive (or zero). We write it with bars: |−7| = 7 and |7| = 7. Distance doesn't care which direction you went. Comparing numbers is different: it's about position on the number line, where farther right means greater.

🧩 Think of it like… the odometer in a car. It only counts how far you've driven, never which direction. Drive 7 miles east or 7 miles west and the trip-counter reads 7 either way — that "7" is the absolute value, the size of the trip with the direction thrown away.
Where it breaks: because the odometer forgets direction, you can't run it backwards — seeing "7" can't tell you whether you headed east (+7) or west (−7). In the same way, once you know |x| = 7, x could be either 7 or −7. Absolute value answers "how far," never "which way."

2 Key terms

Absolute value |x|
The distance of x from 0 — never negative.
Magnitude
Another word for "size" — what absolute value measures.
Inequality (<, >)
A statement of which value is greater (−3 > −8).
Order
Numbers from least to greatest = left to right on the number line.

3 Real-life examples

  • Debt size: a −$30 balance has absolute value $30 — that's how much is owed.
  • Temperature: −10° is colder than −2°, so −10 < −2 — even though |−10| > |−2|.
  • Elevation: a diver at −40 m is deeper than one at −15 m: |−40| > |−15|.
🤔 Pause & think: One diver is at −8 m, another at −30 m. Who is deeper? And whose position is the greater number? Can the same diver win both?
Reveal the thinking
The diver at −30 m is deeper, because the distance from the surface is larger: |−30| = 30 > |−8| = 8. But −30 is the smaller number — it sits farther left on the line, so −30 < −8. So no, the same diver can't win both: deeper means bigger absolute value yet a smaller number. That's exactly why −8 < −3 while |−8| > |−3|.

4 Common doubts

Is absolute value always positive?

Yes — or zero. It's a distance, and distance is never negative.

What's the difference between |−7| and −7?

|−7| = 7 (its distance from 0). −7 is the number itself.

Does a bigger absolute value mean a bigger number?

No! −8 has a bigger absolute value than −3, but −8 is smaller (farther left, colder).

How do I compare two negatives?

Farther left = smaller. So −8 < −3.

5 Step-by-step

  1. Absolute value: measure the distance from 0 (just drop the sign): |−4| = 4.
  2. Compare values: place both on the number line — left is less.
  3. Don't confuse "greater" (position) with "bigger magnitude" (absolute value).

📊 See it · |−4| is a distance

−5 −4 −3 −2 −1 0 1 2 3 4 5 |−4| = 4 units

From −4 to 0 is 4 units, so |−4| = 4. (And |4| = 4 too — same distance.)

✅ Check yourself
  1. True or false: |−8| > |−3|, and −8 > −3.
    answer First is true (8 > 3). Second is false — −8 < −3, since −8 is farther left. Bigger distance, smaller number.
  2. Fill in < or >: −6 ___ −1, and |−6| ___ |−1|.
    answer −6 < −1 (farther left = less), but |−6| > |−1| because 6 > 1.
⚡ Quick recap. Absolute value = distance from 0, always ≥ 0. Comparing is about position: farther right = greater. Don't mix them up — −8 < −3, yet |−8| > |−3|.

Grounded in CA CCSS-M, Grade 6 · 6.NS.7 (absolute value, ordering rational numbers), California Department of Education. Hero image generated with Gemini Nano Banana Pro.