Is absolute value just removing the negative sign?

That shortcut works only when the bars wrap a single plain number, like |-8| = 8. It fails whenever a sign sits outside the bars or an operation sits inside. For example -|-8| = -8, and |5 - 12| = |-7| = 7, not 17. Thinking "distance from zero" instead of "drop the sign" keeps you correct in every case.

Can an absolute value ever be negative?

No. Absolute value reports a distance, and distance is never negative, so |x| ≥ 0 for every number x. An expression like -|6| can equal a negative number, but that minus sign is outside the bars and is applied after the distance is found. There is no number whose absolute value is -6.

Why is -20 less than -3 when 20 is bigger than 3?

Order on the number line depends on position, not on the size of the digits. Both numbers sit to the left of zero, and -20 sits farther left, so it is less. Think of temperatures: 20 degrees below zero is colder than 3 degrees below zero. Among negative numbers, a larger absolute value means a smaller number.

What is the difference between the opposite of a number and its absolute value?

The opposite flips a number to the other side of zero, keeping the distance the same: the opposite of 7 is -7, and the opposite of -7 is 7. Absolute value reports the distance itself and is never negative: |7| = 7 and |-7| = 7. The two agree for negative inputs but differ for positive ones.

M7MATH-1.1

Integers & Absolute Value

Grade 7 guide to integers and absolute value: plot negatives on a number line, find opposites, and read absolute value as distance from zero.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Integers & Absolute Value, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Temperatures below zero, elevations under sea level, yards lost on a football play, money owed — the world is full of quantities that go in two opposite directions. Positive numbers alone cannot describe them, so mathematicians extend the counting numbers backward past zero to create the integers.

In this lesson you will build a number line that runs both ways from zero, learn what it means for two numbers to be opposites, and see why absolute value measures distance rather than direction. Getting this straight now matters: almost every rule you will meet later about adding, subtracting, multiplying, and dividing signed numbers is written in the language of absolute value. Students who think of absolute value as "just drop the negative sign" run into trouble fast. Students who think of it as "how far from zero" almost never do.

What Integers Are and Where They Live on the Number Line

The integers are the whole numbers together with their negatives:

\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots

. There are no fractions or decimals in this set. The number

-4

is an integer;

-4.5

and

\frac{1}{2}

are not (they are rational numbers, which the next lesson takes up).

A number line for integers is a straight line with zero in the middle, positive integers marching right, and negative integers marching left. The tick marks must be evenly spaced, and the spacing has to match on both sides of zero — if one unit is a centimeter to the right, one unit is a centimeter to the left too.

To plot a number like

-6

, start at zero and count six units in the negative direction. Notice what the position tells you: the sign tells you which side of zero you are on, and the digits tell you how far.

A key consequence trips up a lot of students. On the number line, farther right always means greater. So

-8 < -3

, even though

8 > 3

. When you owe eight dollars you are worse off than when you owe three, and

-8

sits farther left. A useful check: pick the two numbers, imagine them as temperatures, and ask which one is colder. Colder is smaller.

Zero is special. It is an integer, it is neither positive nor negative, and it is the reference point everything else is measured from.

Number	Direction from 0	Units from 0
$7$	right (positive)	7
$-7$	left (negative)	7
$-2$	left (negative)	2
$0$	neither	0

Opposites: Same Distance, Different Direction

Two numbers are opposites if they are the same distance from zero but on opposite sides of it. The opposite of

5

-5

; the opposite of

-11

11

. On the number line, opposites are mirror images across zero — fold the line at zero and they land on top of each other.

The negative sign is how we write "the opposite of." That reading is what makes double negatives make sense. The expression

-(-9)

says "the opposite of the opposite of nine." Go left nine, then reverse direction, and you are back at

9

. So

-(-9) = 9

.

Zero is its own opposite, because

-0 = 0

. Zero is the only integer with that property, since every other number sits some positive distance to one side.

Opposites also explain a fact you will use constantly in the next unit: a number plus its opposite is always zero.

6 + (-6) = 0

-40 + 40 = 0

. Walk six units right, then six units left, and you are standing where you started. Pairs like these are called additive inverses.

A warning about vocabulary. "Opposite" and "absolute value" are not the same operation, and mixing them up is one of the most common errors in this unit. The opposite of

-3

3

, and the absolute value of

-3

is also

3

— so far the answers agree. But the opposite of

8

-8

, while the absolute value of

8

8

. For positive inputs the two ideas give different answers, so you have to know which one the problem is asking for.

Absolute Value as Distance from Zero

The absolute value of a number is its distance from zero on the number line. We write it with vertical bars:

|{-7}| = 7

and

|7| = 7

.

Because distance is never negative, absolute value is never negative.

|{-100}| = 100

|0| = 0

, and there is no number whose absolute value is

-4

. If your work ever produces

|x| = -4

with nothing else going on, something has gone wrong.

Think about walking. If a friend says "I walked 3 blocks," that number does not tell you whether they went north or south. Absolute value strips away direction and reports only size, or magnitude.

The rule students memorize — "absolute value makes it positive" — works only when the bars are wrapped around a plain number. It breaks down as soon as a negative sign sits outside the bars. Compare carefully:

Expression	Meaning	Value
$\\|{-6}\\|$	distance of $-6$ from 0	$6$
$-\\|{-6}\\|$	the opposite of that distance	$-6$
$-\\|6\\|$	the opposite of the distance of 6 from 0	$-6$
$\\|{-6}\\| + \\|{-2}\\|$	$6 + 2$	$8$

The bars also act as grouping symbols. In

|3 - 10|

, you simplify inside first to get

|{-7}|

, then take the absolute value to get

7

. A frequent mistake is turning the inside terms positive one at a time and writing

|3 + 10| = 13

. Do the arithmetic inside the bars exactly as written, then measure.

One more contrast worth memorizing: two different numbers can share an absolute value.

|5| = |{-5}| = 5

. So if

|x| = 5

, then

x

could be

5

-5

Using Signed Numbers in Real Situations

Signed numbers describe quantities that move in two opposite directions from an agreed zero point. Choosing that zero point is part of the modeling.

Situation	Zero means	Positive	Negative
Temperature	0 degrees	above zero	below zero
Elevation	sea level	above sea level	below sea level
Bank account	empty account	deposit	withdrawal or debt
Football play	line of scrimmage	yards gained	yards lost
Time before launch	liftoff	after liftoff	before liftoff

In these contexts, the two ideas from this lesson answer two genuinely different questions. The signed number answers "which way and how much?" The absolute value answers "how much, regardless of direction?"

Here is where careful reading matters. A submarine at

-250

feet and a drone at

180

feet: which is farther from sea level? Compare absolute values,

250

against

180

, so the submarine is farther. But which is higher? Compare the signed numbers, and

180 > -250

, so the drone is higher. Same two objects, two different comparisons, two different tools.

Another place students slip: a debt of 60 dollars recorded as

-60

is a larger debt than one recorded as

-25

, but it is the smaller number. Both statements are true at once. "Larger debt" is a statement about absolute value; "smaller number" is a statement about position on the number line. When a word problem sounds contradictory, ask yourself which of the two it is really asking about.

Distance between two points on a number line also uses absolute value: the distance from

-4

3

|3 - (-4)| = |7| = 7

units, which you can verify by counting tick marks.

Comparing, Ordering, and Common Slip-Ups

To order integers, place them mentally on the number line and read left to right. Least is farthest left.

Order

-9, 4, -2, 0, -15, 7

from least to greatest:

-15, -9, -2, 0, 4, 7

. Notice that among the negatives, the one with the largest absolute value is the smallest number. That inversion is the single most common source of errors in this topic.

A reliable two-step method when negatives are involved: first sort by sign (all negatives come before zero, which comes before all positives), then within the negatives order by absolute value from largest to smallest, and within the positives from smallest to largest.

Three errors to watch for.

First, writing

-3 > -8

as though the digits alone decide. Check on a number line:

-8

is farther left, so

-8 < -3

.

Second, dropping the bars too early. In

|2 - 9|

, do the subtraction first:

|{-7}| = 7

.

Third, answering with a negative absolute value. Distance cannot be negative, so

|x| \ge 0

always.

When a problem uses symbols, translate them into words before you compute.

|{-12}| < |{-20}|

reads "twelve units from zero is less than twenty units from zero," which is clearly true. But

-12 < -20

reads "negative twelve is to the left of negative twenty," which is false. Saying the sentence out loud catches the mistake almost every time.

Key terms

Integer.: Any whole number, its opposite, or zero: $\ldots, -2, -1, 0, 1, 2, \ldots$ . Integers include no fractions or decimals.
Number line.: A line with evenly spaced tick marks and zero marked, where positive numbers lie to the right of zero and negative numbers to the left.
Opposites.: Two numbers that are the same distance from zero but on opposite sides, such as $-6$ and $6$ . Zero is its own opposite.
Absolute value.: The distance a number sits from zero on the number line, written with bars: $|{-9}| = 9$ . Absolute value is never negative.
Magnitude.: The size of a quantity without regard to direction or sign; the absolute value of the number describing it.
Additive inverse.: The opposite of a number; adding a number to its additive inverse gives zero, as in $15 + (-15) = 0$ .
Negative number.: A number less than zero, located to the left of zero on the number line and written with a minus sign.
Ordering.: Arranging numbers by position on the number line; a number farther left is always less than a number farther right.

Worked example

A weather station records these midnight temperatures in degrees Celsius over five nights:

-6

3

-11

0

, and

-2

. (a) Plot the temperatures on a number line. (b) Order them from coldest to warmest. (c) Which night's temperature was farthest from freezing (0 degrees), and how far? (d) Write the opposite of each temperature.

(a) Plot. Draw a line and mark zero in the middle, with tick marks at each integer from

-12

4

. Count left from zero for the negatives:

-6

lands six ticks left,

-11

lands eleven ticks left,

-2

lands two ticks left. Count right for

3

. Put

0

at the center mark.

(b) Order. Read the plotted points from left to right. The negatives come first, and among them the one farthest left is smallest. That gives

-11, -6, -2, 0, 3

. Check with the sign-then-size method: negatives sorted by absolute value largest to smallest are

-11

(11 units),

-6

(6 units),

-2

(2 units); then

0

; then

3

.

(c) Farthest from zero. "Farthest from zero" is a question about distance, so take absolute values:

|{-6}| = 6

|3| = 3

|{-11}| = 11

|0| = 0

|{-2}| = 2

. The largest distance is

11

, so the night at

-11

degrees was farthest from freezing, by 11 degrees. Notice that

-11

is the coldest and the farthest from zero — but it is the least number, not the greatest.

(d) Opposites. Flip each across zero: the opposite of

-6

6

; of

3

-3

; of

-11

11

; of

0

0

; of

-2

2

Practice questions

Which statement is true?

$-14 > -5$ because $14 > 5$
$|{-14}| < |{-5}|$ because $-14 < -5$
$-14 < -5$ and $|{-14}| > |{-5}|$
$-|{-14}| = 14$

Answer: $-14 < -5$ and $|{-14}| > |{-5}|$

Position and distance are two separate comparisons. On the number line

-14

sits farther left than

-5

, so

-14 < -5

. But

-14

is 14 units from zero while

-5

is only 5 units from zero, so

|{-14}| = 14 > 5 = |{-5}|

. Both parts of this statement are true at the same time. The first option confuses digit size with position; the second reverses the distance comparison; the last one forgets that the minus sign outside the bars takes the opposite of the distance, giving

-|{-14}| = -14

Evaluate

|4 - 11| + -|{-3}|

and explain each step in words.

Answer: $4$

Absolute value bars group like parentheses, so simplify inside first:

4 - 11 = -7

, giving

|{-7}| = 7

. Next handle

-|{-3}|

. Inside, the distance of

-3

from zero is

3

, and the minus sign outside takes the opposite of that, so

-|{-3}| = -3

. Now add:

7 + (-3) = 4

. The two mistakes to avoid are turning

4 - 11

into

4 + 11

before applying the bars, and letting the bars in

-|{-3}|

cancel the outside minus sign.

A hiker stands at an elevation of

-35

meters relative to sea level. A bird flies at

20

meters. Which is farther from sea level, and which is at the greater elevation? Justify both answers.

Answer: The hiker is farther from sea level; the bird is at the greater elevation.

"Farther from sea level" asks for distance from zero, so compare absolute values:

|{-35}| = 35

and

|20| = 20

. Since

35 > 20

, the hiker is farther from sea level. "Greater elevation" asks about position on the number line, so compare the signed numbers directly:

20 > -35

, so the bird is higher. The same pair of numbers answers the two questions differently, which is exactly why absolute value and signed comparison are separate tools.

FAQ

Is absolute value just removing the negative sign?: That shortcut works only when the bars wrap a single plain number, like $|{-8}| = 8$ . It fails whenever a sign sits outside the bars or an operation sits inside. For example $-|{-8}| = -8$ , and $|5 - 12| = |{-7}| = 7$ , not $17$ . Thinking "distance from zero" instead of "drop the sign" keeps you correct in every case.
Can an absolute value ever be negative?: No. Absolute value reports a distance, and distance is never negative, so $|x| \ge 0$ for every number $x$ . An expression like $-|6|$ can equal a negative number, but that minus sign is outside the bars and is applied after the distance is found. There is no number whose absolute value is $-6$ .
Why is $-20$ less than $-3$ when 20 is bigger than 3?: Order on the number line depends on position, not on the size of the digits. Both numbers sit to the left of zero, and $-20$ sits farther left, so it is less. Think of temperatures: 20 degrees below zero is colder than 3 degrees below zero. Among negative numbers, a larger absolute value means a smaller number.
What is the difference between the opposite of a number and its absolute value?: The opposite flips a number to the other side of zero, keeping the distance the same: the opposite of $7$ is $-7$ , and the opposite of $-7$ is $7$ . Absolute value reports the distance itself and is never negative: $|7| = 7$ and $|{-7}| = 7$ . The two agree for negative inputs but differ for positive ones.

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