Integers in Real Life and Comparing with Absolute Value | Grade 7 Math | LevelKite

A bird flies 12 meters above the sea, and a diver swims 12 meters below the surface. Both are 12 meters from the surface, yet they are in very different places. The number 12 alone cannot tell them apart.

Integers can. The integers are the numbers in the list $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ , which goes on forever in both directions. A positive integer is greater than 0, and a negative integer is less than 0. Call the sea surface 0 and count upward as positive. Then the bird is at $12$ meters and the diver is at $-12$ meters.

Each of these integers carries two facts. Its sign, positive or negative, tells you the direction from 0. Its absolute value tells you how far from 0 it is. For any integer $a$ , the absolute value $\lvert a \rvert$ is the distance from $a$ to 0, so it is never negative. Here $\lvert 12 \rvert = 12$ and $\lvert -12 \rvert = 12$ , because the bird and the diver are both 12 meters from the surface.

In this lesson you will write real-world quantities as integers, and then use absolute value to compare integers.

To write a real-world quantity as an integer, settle three things first.

What does 0 mean? Choose the reference point, the value you measure from, such as sea level or an account balance of zero.
Which direction is positive? Words such as above, rise, gain and deposit usually point in the positive direction. Words such as below, drop, loss and owe then point in the negative direction.
What is the unit? Keep it with the number, because $-5$ meters and $-5$ dollars describe different things.

Now write how far the quantity is from the reference point, and put a negative sign in front if it lies in the negative direction. A positive integer needs no sign: $+12$ and $12$ are the same number. A quantity exactly at the reference point is $0$ , which is neither positive nor negative.

An integer can also describe a change. Then 0 means no change. With a rise counted as positive, a rise of 5 degrees is $5$ and a drop of 5 degrees is $-5$ .

Six real-world quantities written as integers

Situation	What 0 means	Positive direction	Integer with unit
A submarine is 150 meters below sea level.	Sea level	Above sea level	$-150$ meters
The temperature is 4 degrees Celsius below zero.	0 degrees Celsius	Above 0 degrees	$-4$ degrees Celsius
You have 25 dollars in your account.	No money and no debt	Money you have	$25$ dollars
A player loses 30 points in one turn.	No change in the score	Points gained	$-30$ points
An elevator goes up 3 floors.	No change in floor	Floors up	$3$ floors
It is 10 seconds before a rocket launch.	The moment of launch	Time after launch	$-10$ seconds

Worked example

Writing positions as integers

A map of a coastal park measures every height from sea level. Write an integer, with its unit, for each place: a lookout point 240 meters above sea level, a shipwreck 35 meters below sea level, and a beach exactly at sea level.

State what 0 means and which direction is positive. Let 0 meters be sea level, and let positive mean above sea level. Then negative means below sea level.
The lookout point is above sea level, which is the positive direction, and it is 240 meters from sea level. So its integer is positive with absolute value 240.
$240 \text{ meters}$
The shipwreck is below sea level, which is the negative direction, and it is 35 meters from sea level. So its integer is negative with absolute value 35.
$-35 \text{ meters}$
The beach is exactly at the reference point, so its distance from sea level is 0 meters. Zero is neither positive nor negative.
$0 \text{ meters}$
Check each integer against the situation. Its absolute value must equal the stated distance from sea level. A place above sea level must be positive, a place below must be negative, and a place at sea level must be 0.
$\lvert 240 \rvert = 240, \quad \lvert -35 \rvert = 35, \quad \lvert 0 \rvert = 0$

Answer

With sea level as $0$ and upward as positive, the lookout point is at $240$ meters, the shipwreck is at $-35$ meters and the beach is at $0$ meters.

Now compare two integers. On the number line, every integer has its own point, and the integers increase from left to right. So of two different integers, the one farther left is the lesser one. The symbols $<$ (is less than) and $>$ (is greater than) record this order.

Absolute value answers a different question: how far from 0? So each pair of integers comes with two separate questions.

Which is greater? That depends on position on the number line.
Which is farther from 0? That depends on absolute value.

Take two temperatures, $-9$ and $4$ degrees Celsius. The integer $-9$ is negative, so it is left of 0, and $4$ is positive, so it is right of 0. That makes $-9$ the lesser integer, the colder temperature. But $\lvert -9 \rvert = 9$ and $\lvert 4 \rvert = 4$ , and $9 > 4$ , so $-9$ is farther from 0. The two questions have different answers:

-9 < 4 \qquad \text{but} \qquad \lvert -9 \rvert > \lvert 4 \rvert

Key concept

Sign gives the side, absolute value gives the distance

A positive or negative integer tells you two things. Its sign tells you which side of 0 it is on. Its absolute value tells you how far from 0 it is, and that distance is never negative.

Order follows position on the number line, not distance from 0. So absolute value alone cannot tell you which of two different integers is greater. First use the signs to find each integer's side of 0. If both are on the same side, the absolute values then show which one is farther from 0 on that side.

Comparing two different integers with absolute value

The two integers are	The greater integer is	Reason	Example
both positive	the one with the greater absolute value	Right of 0, farther from 0 means farther right.	$\lvert 8 \rvert > \lvert 3 \rvert$ , so $8 > 3$
both negative	the one with the smaller absolute value	Left of 0, farther from 0 means farther left.	$\lvert -8 \rvert > \lvert -3 \rvert$ , so $-8 < -3$
one positive, one negative	the positive one, whatever the absolute values are	Every negative integer is left of 0 and every positive integer is right of 0.	$-8 < 3$ , $-3 < 8$ and $-8 < 8$
0 and one other integer	0 if the other is negative; the other integer if it is positive	0 is right of every negative integer and left of every positive integer.	$-8 < 0$ and $0 < 3$

Worked example

Comparing two negative integers

Use absolute value to decide which integer is greater, $-15$ or $-9$ .

Look at the signs first, because the sign tells you which side of 0 an integer is on. Both integers are negative, so both are to the left of 0.
$-15 < 0 \quad \text{and} \quad {-9} < 0$
Find each absolute value, which is the distance from 0. Negative 15 is 15 units from 0, and negative 9 is 9 units from 0.
$\lvert -15 \rvert = 15 \quad \text{and} \quad \lvert -9 \rvert = 9$
Compare the two distances. Since 15 is greater than 9, negative 15 is farther from 0 than negative 9.
$15 > 9, \quad \text{so} \quad \lvert -15 \rvert > \lvert -9 \rvert$
Turn distance into position. Both integers are to the left of 0, so the one farther from 0 is the one farther left. Farther left on the number line means less.
$-15 < -9$

Answer

The greater integer is $-9$ : $-15 < -9$ , even though $\lvert -15 \rvert > \lvert -9 \rvert$ .

Worked example

Lowest balance, largest debt

A school store keeps an account for each student. The balance is the amount in the account. Amara owes the store 12 dollars, Ben owes the store 7 dollars, and Chen has 5 dollars in the account. Write each balance as an integer. Then order the balances from least to greatest and decide who owes the most.

Set the reference point and the positive direction. Let 0 dollars be an account with no money and no debt, and let positive mean money in the account. Then money owed is negative. In dollars, Amara's balance is negative 12, Ben's is negative 7 and Chen's is 5.
$-12, \quad -7, \quad 5$
Use the signs first. Chen's balance is positive and the other two are negative. Every positive integer is to the right of 0 and every negative integer is to the left of 0, so Chen's balance is the greatest.
$-12 < 5 \quad \text{and} \quad {-7} < 5$
The signs cannot separate two negative balances, so use absolute value. Find each distance from 0, then compare the distances.
$\lvert -12 \rvert = 12, \quad \lvert -7 \rvert = 7, \quad 12 > 7$
Both of these balances are to the left of 0, so the one farther from 0 is farther left and is the lesser one. Now all three balances can go in order.
$-12 < -7 < 5$
Answer the debt question with absolute value. The size of a debt is the distance of a negative balance from 0. Amara's debt is 12 dollars and Ben's debt is 7 dollars. Chen's balance is positive, so Chen owes nothing.
$\lvert -12 \rvert > \lvert -7 \rvert$

Answer

From least to greatest, the balances are $-12$ , $-7$ and $5$ dollars. Amara has the lowest balance and also the largest debt, because $-12 < -7$ while $\lvert -12 \rvert > \lvert -7 \rvert$ .

Takeaway. A positive or negative integer packs two facts into one number: its sign gives the direction from the reference point, and its absolute value gives the distance. To represent a real-world quantity, say what 0 means, say which direction is positive, and keep the unit.

To compare integers, remember that order follows position on the number line, not distance from 0. Check the signs first. For two positive integers, the greater absolute value belongs to the greater integer. For two negative integers, the greater absolute value belongs to the lesser integer. For one of each, the positive integer is the greater one.