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Inequalities on
Number Lines

Learn how the symbols \( < \), \( > \), \( \leq \), and \( \geq \) describe where numbers sit on a number line and what that means about their size.

6th Grade Math

What Is an Inequality?

An inequality is a mathematical statement that compares two values and tells us they are not necessarily equal. Instead of using \( = \), we use symbols like \( < \), \( > \), \( \leq \), and \( \geq \).

The Four Inequality Symbols

\( < \) Less Than
Left number is smaller. Example: \( 2 < 5 \)
\( > \) Greater Than
Left number is larger. Example: \( 9 > 4 \)
\( \leq \) Less Than or Equal
Left number is smaller or the same. Example: \( 3 \leq 3 \)
\( \geq \) Greater Than or Equal
Left number is larger or the same. Example: \( 7 \geq 5 \)

Position on the Number Line

0 1 2 3 4 5 6 2 5 2 is LEFT of 5
So \( 2 < 5 \)

The Core Rule

Left = Less Than
If \( a \) is to the left of \( b \) on the number line, then \( a < b \).
Right = Greater Than
If \( a \) is to the right of \( b \) on the number line, then \( a > b \).

Negative Numbers on the Line

-4 -3 -2 -1 0 1 2 3 -3 1 -3 is LEFT of 1
So \( -3 < 1 \)

\( < \) vs \( \leq \): What's the Difference?

\( x < 4 \) (strict)
 0 1 2 3 4 5 Open: 4 NOT included
\( x \leq 4 \) (inclusive)
 0 1 2 3 4 5 Closed: 4 IS included

Reading Two Numbers on a Line

When you see two numbers plotted on a number line, you can instantly write an inequality. Here is the method:

1
Locate both numbers on the number line.
2
Identify which is further left. That number is the smaller one.
3
Write the inequality. The left number goes on the left with \( < \), or the right number goes on the left with \( > \).
4
Check: Does the symbol's opening point toward the bigger number? If yes, you're correct.

Step-by-Step Strategy

1
Draw or picture the number line in your head, with negative numbers on the left and positive on the right.
2
Plot both numbers. Even a rough sketch helps you see which is further left.
3
Compare positions. Left = less than. Right = greater than.
4
Write the inequality using \( < \) or \( > \). Remember: the symbol always "opens" toward the bigger number.
5
Translate to words: "\( -2 < 3 \)" means "negative two is less than three" or equivalently "negative two is to the left of three on the number line."

Example 1: Comparing Integers

-5 -4 -3 -2 -1 0 1 2 3 4 5 -4 2
\( -4 < 2 \) because -4 is LEFT of 2

Example 2: Two Negatives

-6 -5 -4 -3 -2 -1 0 -5 -1
\( -5 < -1 \) because -5 is LEFT of -1

Example 3: Fractions and Decimals

0 0.5 1 1.5 2 2.5
\( \frac{3}{4} \)
 2.3
\( \frac{3}{4} \) is LEFT of 2.3, so \( \frac{3}{4} < 2.3 \)

Why Does Position Equal Value?

The number line is built so that each step to the right adds the same amount. Moving right always means adding a positive value, which makes the number bigger. Moving left means subtracting, making it smaller.

Move Right
Add a positive amount. The number increases.
Move Left
Subtract (or add a negative). The number decreases.

Flipping the Inequality

Every inequality can be read in two directions. If you swap the two numbers, you must also flip the symbol:

\( 3 < 7 \)
"3 is less than 7"
\( 7 > 3 \)
"7 is greater than 3"

Common Mistakes to Avoid

✕ Ignoring the Negative Sign
Thinking \(-7 > 2\) because 7 is "bigger" than 2. Always check position on the number line: \(-7\) is far left, so \(-7 < 2\).
✕ Confusing Open and Closed Circles
Using a closed circle for \( < \) or an open circle for \( \leq \). Remember: open = not included (strict), closed = included.
✕ Flipping Without Swapping
Writing \( 3 > 7 \) when you mean \( 3 < 7 \). If the small number is on the left, the symbol must point left (toward it).

Summary Table

Symbol Meaning Number Line Circle Type
\( < \) Less than Left of Open
\( > \) Greater than Right of Open
\( \leq \) Less than or equal to Left of or same spot Closed
\( \geq \) Greater than or equal to Right of or same spot Closed

Question 1 of 5

On a number line, \(-3\) is to the left of \(4\). Which inequality is correct?

Question 2 of 5

Which statement correctly describes the positions of \(-8\) and \(-2\) on a number line?

Question 3 of 5

If \( a > b \), what does this tell you about the positions of \( a \) and \( b \) on a number line?

Question 4 of 5

A number line shows a closed circle at \(5\) with shading to the left. Which inequality does this represent?

Question 5 of 5

\(-1.5\) and \(\frac{1}{2}\) are plotted on a number line. Which two statements are both true?