Ordering Integers, Opposites, and Absolute Value | Grade 7 Math | LevelKite

You already know how to graph integers on a number line. Integers are the counting numbers ( $1, 2, 3$ and so on), their negatives ( $-1, -2, -3$ and so on), and $0$ . That same line can tell you which of two integers is less.

On the number lines in this lesson, each tick mark is $1$ more than the tick on its left, so numbers increase as you move right. The space between two neighboring ticks is $1$ unit. Negative integers sit to the left of $0$ , and positive integers sit to the right.

So position tells you order. For any two different integers, the one farther left is less and the one farther right is greater. The symbol $<$ means is less than, and $>$ means is greater than.

The number line below marks $-7$ and $-3$ . Which one do you think is greater? The first example finds out.

Markers sit on the ticks for

-7

and

-3

. Both ticks are to the left of

0

.

Worked example

Compare two negative integers

Which is greater, $-7$ or $-3$ ? Write the comparison with $<$ or $>$ . The number line above has a marker on the tick for each number.

Locate -7. It is negative, so start at 0 and count 7 units to the left. You land on the tick that holds the marker labeled -7.
Locate -3 the same way. Start at 0 and count 3 units to the left. You land on the tick that holds the marker labeled -3.
Compare the two positions. Both markers are left of 0, but -7 is 7 units from 0 and -3 is only 3 units from 0, so -7 is farther left. On a number line where numbers increase to the right, the number farther left is less.
$-7 < -3$
The question asks which is greater. Saying that -7 is less than -3 means the same as saying that -3 is greater than -7, so write the comparison with the greater number first.
$-3 > -7$

Answer

The greater number is $-3$ . You can write this as $-3 > -7$ or as $-7 < -3$ .

Five markers sit on the ticks for

-6

,

-2

,

0

,

3

and

7

, the integers in the next example.

Worked example

Order five integers

Order $3, -6, 7, 0, -2$ from least to greatest, and then from greatest to least. The number line above has a marker on the tick for each of these integers.

Sort the numbers by their side of 0. The negative integers -6 and -2 are left of 0, the integer 0 is at 0 itself, and the positive integers 3 and 7 are right of 0. Every number left of 0 is less than 0, and 0 is less than every number right of 0, so from least to greatest the negatives come first and the positives come last.
Order the two negative integers. Counting from 0, -6 is 6 units to the left and -2 is 2 units to the left. So -6 is farther left, which makes it the lesser one.
$-6 < -2$
Order the two positive integers. Counting from 0, 3 is 3 units to the right and 7 is 7 units to the right. So 7 is farther right, which makes it the greater one.
$3 < 7$
Join the pieces by reading the five markers from left to right. Left to right is least to greatest, because each tick is 1 more than the tick on its left.
$-6 < -2 < 0 < 3 < 7$
For greatest to least, read the same markers from right to left. Each tick is 1 less than the tick on its right, so each number you read is less than the one before it.
$7 > 3 > 0 > -2 > -6$

Answer

From least to greatest, the integers are $-6, -2, 0, 3, 7$ . From greatest to least, they are $7, 3, 0, -2, -6$ .

Position answers the question which is less? The same line answers two more questions: how far is a number from $0$ , and what sits the same distance away on the other side? Let $a$ stand for any integer.

You already know the absolute value of $a$ , which you write as $|a|$ . It is the distance from $a$ to $0$ , and you count it in units. A distance is never negative, so $|a|$ is never negative.
The opposite of $a$ , which you write as $-a$ , is the number that is the same distance from $0$ as $a$ but on the other side of $0$ . The opposite of $0$ is $0$ , because the only number that is $0$ units from $0$ is $0$ itself.

The number line below has markers at $-5$ , $0$ and $5$ . Count from the marker at $0$ : the marker at $-5$ is $5$ units to the left, and the marker at $5$ is $5$ units to the right. They are the same distance from $0$ on different sides, so $-5$ and $5$ are opposites of each other. Each one is $5$ units from $0$ , so $|-5| = 5$ and $|5| = 5$ .

Counting from

0

, the marker at

-5

is

5

units to the left and the marker at

5

is

5

units to the right.

Key concept

Position gives order; distance from 0 gives absolute value

Every integer $a$ has a position on the number line and a distance from $0$ . Each one answers a different question.

Order follows position, not magnitude (distance from $0$ ). On a line where numbers increase to the right, of two different integers the one farther left is less.
Absolute value $|a|$ is the distance from $0$ . It is never negative, and it is $0$ only when $a = 0$ .
For $a$ other than $0$ , the opposite $-a$ is the same distance from $0$ as $a$ , on the other side of $0$ . The opposite of $0$ is $0$ . So $|a| = |-a|$ for every integer $a$ .

In general, the distance between two numbers $p$ and $q$ is $|p - q|$ . Choosing $q = 0$ gives $|p - 0| = |p|$ , the distance from $0$ .

The last two columns match for the negative integer and for zero, but not for the positive integer.

Integer	Side and distance	Absolute value	Opposite
$9$	right of $0$ , $9$ units away	$9$	$-9$
$-9$	left of $0$ , $9$ units away	$9$	$9$
$0$	at $0$ , $0$ units away	$0$	$0$

Worked example

Find an absolute value and an opposite

Let $n = -12$ . Find the absolute value of $n$ , the opposite of $n$ , and the absolute value of that opposite.

Read the side and the distance from the number itself. The integer -12 has a negative sign, so it is left of 0. Without its sign the number is 12, so it is 12 units from 0.
$n = -12$
Find the absolute value of n. By definition it is the distance from -12 to 0, which is 12 units. A distance is never negative, so the result has no negative sign.
$|n| = |-12| = 12$
Find the opposite of n. Keep the distance of 12 units and switch from the left side of 0 to the right side. The number 12 units to the right of 0 is 12. The opposite is positive here because n itself is negative.
$-n = -(-12) = 12$
Find the absolute value of the opposite. The opposite is 12, which is 12 units to the right of 0, so its distance from 0 is 12 units.
$|-n| = |12| = 12$
Check the results against the definitions. The numbers -12 and 12 are on different sides of 0 and the same 12 units from it, so they are opposites, and both absolute values equal that shared distance. The opposite of n equals its absolute value here because n is negative; for a positive integer the opposite is negative, so the two differ.
$|n| = |-n| = 12$

Answer

The absolute value of $-12$ is $12$ , and the opposite of $-12$ is $12$ . The absolute value of that opposite is also $12$ .

Takeaway. An integer's position on the number line gives its order: of two different integers, the one farther left is less, even when it is farther from $0$ . Its distance from $0$ gives its absolute value, which is never negative. Its opposite keeps that distance and switches the side of $0$ , and the opposite of $0$ is $0$ .

Next, you will use positive and negative integers to represent real-world quantities, and you will compare integers using absolute value.