Kid-friendly explanation
The not equal sign looks like this: ≠. It means "is not the same as." If 5 ≠ 3, that says 5 is not the same as 3. You can think of it as a sign that points out when two amounts do not match!
Worked examples with the not equal sign
Example 1 (younger grades): Comparing whole numbers
Compare 347 and 372 using the not equal sign.
Look at the hundreds: both numbers have 3 hundreds. Look at the tens: 347 has 4 tens and 372 has 7 tens, so the tens are different. That means the whole numbers are different. You can write: 347 ≠ 372.
Example 2 (older grades): Comparing fractions and decimals
Compare 3/4 and 0.75, and then compare 2/5 and 0.4, using the not equal sign.
First, 3/4 means 3 parts out of 4 equal parts. As a decimal, 3/4 = 0.75, so these are equal. You write 3/4 = 0.75, not 3/4 ≠ 0.75.
Next, 2/5 means 2 parts out of 5 equal parts. To write 2/5 as a decimal, think of it as 2 ÷ 5. Since 2 ÷ 5 = 0.4, 2/5 = 0.4. Now compare 2/5 and 0.5. Because 0.4 is less than 0.5, they are different values. You write: 2/5 ≠ 0.5.
How the not equal sign is used in later grades
The not equal sign is used any time a comparison shows two values are different. Students use it with whole numbers, fractions, decimals, and rational numbers (numbers that can be written as a fraction a/b where b is not equal to 0).
In algebra, the not equal sign helps describe when two expressions do not match. For example, if x + 3 ≠ 10, then x cannot be 7, because 7 + 3 = 10. It also appears in conditions such as "b is not equal to 0" when writing fractions a/b or unit rates a:b, because dividing by 0 is not allowed.
In more advanced courses, the not equal sign shows up in equations and functions, such as exponential equations y = a b^x where b is greater than 0 and not equal to 1, or when describing linear equations y = mx + b where b is not equal to 0, meaning the line does not pass through the origin.
Common mistakes with the not equal sign
- Mistake: Using the not equal sign when only the way a number looks is different, such as writing 1/2 ≠ 0.5.
Correction: 1/2 and 0.5 represent the same value, so they are equal; the correct statement is 1/2 = 0.5. - Mistake: Forgetting that a fraction a/b requires b ≠ 0 and writing something like 5/0 as if it were a normal number.
Correction: Division by 0 is undefined, so denominators and unit rates must have a nonzero bottom number; b is not equal to 0. - Mistake: Mixing up inequality symbols and writing 6 < 6 or 4 > 4 when trying to say two numbers are different.
Correction: If two numbers are the same, use =. If they are different but you are not comparing which is larger, use the not equal sign: for example, 5 ≠ 7.
How the not equal sign shows up in everyday life
The idea of "not equal" appears whenever two amounts do not match. If one person has 3 slices of pizza and another has 5 slices, the amounts are not equal. If a game score is 10 to 8, the scores are not equal. The not equal sign is the math way to show that mismatch.