Comparing Integers Using <, >, and =

Welcome back. Last lesson focused on absolute value: the distance of an integer from zero. That number-line idea now becomes the main tool for deciding which of two integers has the greater value.

In this lesson, you will compare integers using $<$, $>$, and $=$. You will learn how number-line position settles every comparison, including the sometimes confusing case of comparing two negative integers.

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The number-line rule

A number line is arranged so that values increase as you move to the right and decrease as you move to the left.

That gives one rule that works for every pair of integers:

The integer farther right is greater. The integer farther left is less.

In the image, $-3$ is to the left of $2$. Therefore,

$$-3<2$$

Read this aloud as “negative three is less than two.”

The sign also works in the reverse direction:

$$2>-3$$

Read this as “two is greater than negative three.” These two statements communicate exactly the same comparison.

A quick visual rule for the symbols:

Symbol	Meaning	How to read it
$<$	less than	“is less than”
$>$	greater than	“is greater than”
$=$	equal to	“is equal to”

For $<$ and $>$, the wide, open side faces the greater number. The narrow point faces the smaller number. But rely first on the meaning of the statement, not only on memorizing the shape.

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Use the symbols to tell a true statement

When you compare two integers, pay attention to the order in which they are written. You are filling in a statement that must be true from left to right.

Suppose you must complete:

$$-8\square 4$$

On a number line, $-8$ is left of $4$, so $-8$ is less than $4$:

$$-8<4$$

It would be incorrect to write $-8>4$, because that says “negative eight is greater than four.”

Here is a reliable method:

1. Picture or draw the two integers on a number line.
2. Find which integer is farther right.
3. State whether the left-hand number is less than, greater than, or equal to the right-hand number.
4. Write $<$, $>$, or $=$ to make that sentence true.

For example, compare $6$ and $-1$:

• $6$ is farther right.
• The comparison is written with $6$ first.
• So the correct statement is:

$$6>-1$$

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The four comparison situations

The number-line rule always works, but these patterns will help you compare quickly without drawing a full line each time.

Two positive integers

Among positive integers, the number with the larger usual counting value is greater.

$$13>7$$

Since $13$ is farther right than $7$, it is greater.

One negative and one positive integer

Every positive integer is greater than every negative integer. Zero lies between them.

$$-25<3$$

You do not need to compare $25$ and $3$ here. The signs already tell you their positions: a negative is left of zero, while a positive is right of zero.

Zero and another integer

Zero is greater than every negative integer and less than every positive integer.

$$0>-6$$ $$0<11$$

Remember: zero is neither positive nor negative, but it has its own fixed place between them.

Two negative integers

This case deserves extra care. With two negatives, the integer closer to zero is greater because it lies farther right.

Compare $-4$ and $-9$. Since $-4$ is closer to zero, it is farther right:

$$-4>-9$$

A useful check comes from the previous lesson: among two negative integers, the one with the smaller absolute value is greater.

$$|-4|=4$$ $$|-9|=9$$

Because $4$ is the smaller distance from zero, $-4$ is the greater integer. This is only a shortcut for a pair of negative integers; the number-line rule is the main idea.

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Equality: when neither number is greater

Use the equals sign, $=$, when both sides name the same integer.

$$-12=-12$$ $$0=0$$

Equality does not mean that two numbers merely have the same absolute value. For example, $5$ and $-5$ are both five units from zero, but they are at different positions on the number line:

$$-5<5$$

They have equal absolute values, yet they are not equal integers.

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A common negative-number mistake

It is easy to look at $-18$ and $-6$ and think that $18$ is bigger than $6$, so $-18$ must be bigger. But the negative signs reverse that idea on the number line.

$-18$ is much farther left than $-6$, so:

$$-18<-6$$

Think of temperature. A temperature of $-18$ degrees is colder than $-6$ degrees, so $-18$ is the smaller value.

Do not say “the bigger negative wins.” Instead, ask:

Which negative integer is closer to zero?

For $-18$ and $-6$, $-6$ is closer to zero, so it is greater.

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Comparing integers in everyday situations

Integers often represent values relative to a reference point, such as sea level, zero dollars, or zero degrees. First translate the situation into integers, then compare the integer values.

Temperature

At noon, one town has a temperature of $-2$ degrees and another has a temperature of $-11$ degrees.

Since $-2$ is closer to zero and farther right on the number line:

$$-2>-11$$

The first town is warmer.

Elevation

A cave is $35$ meters below sea level, while another cave is $12$ meters below sea level. Their elevations are $-35$ meters and $-12$ meters.

$$-35<-12$$

The cave at $-35$ meters is lower.

Money and the meaning of “bigger”

Suppose one account balance is $-15$ dollars and another is $-25$ dollars:

$$-15>-25$$

The balance $-15$ dollars is greater because it is closer to zero. However, someone owing $25$ dollars has the larger debt amount. This does not change the integer comparison; it shows why you must distinguish between:

• the greater integer, $-15$;
• the larger amount owed, $25$.

Context tells you what the sign means. The number line tells you which integer is greater.

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A fast comparison checklist

Before choosing a symbol, use this checklist:

If you see...	Decide this
Two positives	The larger counting number is greater.
A positive and a negative	The positive integer is greater.
Zero and a positive	The positive integer is greater.
Zero and a negative	Zero is greater.
Two negatives	The one closer to zero is greater.
The same integer twice	Use $=$.

Then reread the completed inequality as words. For example, after writing

$$-30<-8$$

say: “Negative thirty is less than negative eight.” Since that is true, the symbol is correct.

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Key takeaways

To compare any two integers, use the number-line rule: farther right means greater; farther left means less.

• Use $<$ for “less than,” $>$ for “greater than,” and $=$ when the integers are the same.
• Every positive integer is greater than zero and every negative integer.
• Zero is greater than all negative integers and less than all positive integers.
• When comparing two negative integers, the one closer to zero is greater:

$$-3>-10$$

• Absolute value can help with two negatives, but number-line position is the most dependable explanation.

Next, you will extend this skill by arranging several integers in ascending order (least to greatest) or descending order (greatest to least).