Comparing Numbers by Position on a Number Line | Grade 6 Math | LevelKite

Which is greater, $-3$ or $-7$ ? The number line gives a clear answer.

You can already find positive and negative numbers on a number line. In this lesson, you will use those positions to decide which of two numbers is greater. You will also read a comparison written in symbols as a statement about left and right.

Every number line in this lesson is horizontal. Its ticks (the small marks) are equally spaced, so every step from one tick to the next is the same size. Its numbers increase as you move to the right.

Start with what you know. On a number line with steps of $1$ , count up: $3, 4, 5$ . Each count moves one step to the right and lands on a greater number. So $5$ is to the right of $3$ , and $5$ is greater than $3$ .

Now count down from $3$ : $2, 1, 0$ . Keep going past zero: $-1, -2, -3$ . Each count moves one step to the left and lands on a lesser number. The pattern does not turn around at $0$ .

So on a horizontal number line whose numbers increase to the right, greater means farther to the right and less means farther to the left. This is true on both sides of $0$ .

In this lesson, an inequality is a statement that compares two different numbers with one of these symbols:

$<$ means “is less than”
$>$ means “is greater than”

You read an inequality from left to right. $3 < 5$ says “3 is less than 5”. $5 > 3$ says “5 is greater than 3”. Both statements describe the same picture: $3$ on the left and $5$ on the right. In a true inequality, the pointed end of the symbol points at the lesser number.

Key concept

An inequality tells you left and right

Let $a$ and $b$ stand for two different numbers on a horizontal number line whose numbers increase from left to right.

$a < b$ means $a$ is to the left of $b$ .
$a > b$ means $a$ is to the right of $b$ .

This holds for any two different numbers on such a line, on either side of $0$ . So to compare two numbers, find which one is farther right. That one is the greater number.

The markers show

-7

and

-3

. Each step is

1

, and the marker at

-3

is 4 steps to the right of the marker at

-7

.

Worked example

Read an inequality as positions

The number line above has markers (marked points) at $-7$ and $-3$ . What does the inequality $-3 > -7$ tell you about where these two numbers are on the line?

Read the inequality in words. The symbol between the two numbers means “is greater than”, so the statement says that −3 is greater than −7.
$-3 > -7$
Turn “greater” into a position. This number line is horizontal and its numbers increase to the right, so the greater number is the one farther right. That means −3 is to the right of −7.
Check with the markers. Each marker sits exactly on a tick. Neighboring printed numbers, such as −7 and −6, are 1 apart, so each step is 1. Start at the marker at −7 and step right: −6, −5, −4, −3. After 4 equal steps you reach the marker at −3, so −3 is to the right of −7, just as the inequality says.
Describe the same picture starting from the other number. −7 is to the left of −3, and on this line left means less. So −7 is less than −3. Both inequalities are true together.
$-7 < -3$

Answer

The inequality $-3 > -7$ tells you that $-3$ is to the right of $-7$ on the number line. The same positions also give $-7 < -3$ .

An integer is a whole number or the negative of a whole number: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

To compare two integers, find which one is farther right on a horizontal number line whose numbers increase to the right. Counting equal steps of $1$ from $0$ tells you where each integer is. For example, $4$ is 4 steps to the right of $0$ , and $-6$ is 6 steps to the left of $0$ .

Sometimes the side of $0$ settles the comparison. On a horizontal number line whose numbers increase to the right, every negative number is to the left of $0$ and every positive number is to the right of $0$ . So a negative number is always less than $0$ and less than every positive number. For example, $-6 < 0$ and $-6 < 4$ .

It does not matter that $-6$ is more steps from $0$ than $4$ is. The two sets of steps go in opposite directions.

When both integers are negative, both are on the left side of $0$ , so you have to look more closely. The next example shows how.

Worked example

Compare two negative integers

Which is greater, $-8$ or $-5$ ? Write the comparison as an inequality.

Picture a horizontal number line whose numbers increase to the right, with a tick at every integer, so each step is 1. Find −8 by counting equal steps from 0. The negative sign tells you to go left, so −8 is 8 steps to the left of 0.
Find −5 the same way. Its negative sign also sends you left, so −5 is 5 equal steps to the left of 0.
Compare the positions. Both numbers are to the left of 0, and 8 steps reaches farther left than 5 steps of the same size. So −8 is to the left of −5. To check, start at −8 and step right 3 times: −7, −6, −5.
Turn the positions into symbols. On this line the number farther right is the greater one, so −5 is greater than −8.
$-5 > -8$

Answer

$-5$ is greater than $-8$ , because $-5$ is to the right of $-8$ on the number line. As an inequality, $-5 > -8$ .

A rational number is a number you can write as a fraction whose numerator (top number) and denominator (bottom number) are integers, with a denominator that is not $0$ . Integers, fractions such as $-\frac{3}{4}$ , and decimals that end, such as $-1.3$ , are all rational numbers.

Rational numbers are points on the same number line as the integers, so you compare them the same way. On a horizontal number line whose numbers increase to the right, the number farther right is greater. To see both numbers, use a number line whose ticks land exactly on both of them.

The side of $0$ still helps. A negative rational number is always less than $0$ and less than every positive rational number, so $-2.5 < 0.25$ .

Ten equal steps separate the printed numbers

-1

and

0

, so each step is

0.1

. The markers show

-1.3

and

-0.6

.

Worked example

Compare two negative decimals

The number line above has markers at $-1.3$ and $-0.6$ . Which symbol, $<$ or $>$ , belongs between $-1.3$ and $-0.6$ , in that order?

Find the size of one step. The printed numbers −1 and 0 are next to each other and are 1 apart. There are 10 equal steps between them, so each step is one tenth.
$1 \div 10 = 0.1$
Find −0.6 by counting equal steps from 0. Six tenths is 6 steps of 0.1, and the negative sign sends you left. So −0.6 is 6 steps to the left of 0, where its marker sits.
$6 \times 0.1 = 0.6$
Find −1.3 the same way. Thirteen tenths is 13 steps of 0.1 to the left of 0. The first 10 steps take you to −1, and 3 more steps take you to the marker at −1.3.
$13 \times 0.1 = 1.3$
Compare the positions. Both markers are to the left of 0, and 13 steps reaches farther left than 6 steps of the same size. So −1.3 is to the left of −0.6.
Choose the symbol. On this number line the number farther left is the lesser one, so −1.3 is less than −0.6. The less-than symbol has its pointed end toward −1.3.
$-1.3 < -0.6$

Answer

The symbol $<$ belongs between them: $-1.3 < -0.6$ , because $-1.3$ is to the left of $-0.6$ on the number line.

Four equal steps separate the printed numbers

-1

and

0

, so each step is

\frac{1}{4}

. Point A is 3 steps left of

0

and Point B is 2 steps left of

0

.

Worked example

Compare two negative fractions

Compare $-\frac{3}{4}$ and $-\frac{1}{2}$ . On the number line above, Point A and Point B mark these two numbers. Which point is which, and which number is greater?

Find the size of one step. The printed numbers −1 and 0 are next to each other and are 1 apart, with 4 equal steps between them. So each step is one fourth.
$1 \div 4 = \frac{1}{4}$
Find negative three fourths by counting equal steps from 0. Three fourths is 3 steps of one fourth, and the negative sign sends you left. Three steps to the left of 0 is Point A.
$3 \times \frac{1}{4} = \frac{3}{4}$
Rewrite one half in fourths, so both numbers use the same step. Multiply the numerator and the denominator by 2. That is the same as multiplying the fraction by 1, because 2 divided by 2 is 1, so its value does not change. Negative one half and negative two fourths are the same point on the line.
$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$
Find negative two fourths. Two fourths is 2 steps of one fourth, so go 2 steps to the left of 0. That is Point B.
$2 \times \frac{1}{4} = \frac{2}{4}$
Compare the positions. Both points are to the left of 0, and 3 steps reaches farther left than 2 steps of the same size. So Point A is to the left of Point B, and the number at Point B is the greater one.
$-\frac{1}{2} > -\frac{3}{4}$

Answer

Point A is $-\frac{3}{4}$ and Point B is $-\frac{1}{2}$ . Point B is farther right, so $-\frac{1}{2}$ is greater than $-\frac{3}{4}$ . As an inequality, $-\frac{1}{2} > -\frac{3}{4}$ .

Takeaway. On a horizontal number line whose numbers increase to the right, an inequality is a statement about position. For any two different numbers $a$ and $b$ on such a line, $a < b$ says that $a$ is to the left of $b$ , and $a > b$ says that $a$ is to the right of $b$ . To compare two rational numbers, including integers, find each one by counting equal steps from $0$ . The number farther right is the greater one.

Next, you will put several rational numbers in order and write statements of order for real-world situations.