Greater Than Less Than- A Complete Guide to Order of Operations
Greater Than, Less Than, and Order of Operations Explained
These are two fundamental math concepts that confuse people of all ages. You're not alone if you've ever stared at a problem wondering where to start. This guide cuts through the confusion and gives you what actually works.
Understanding Greater Than and Less Than
The symbols > and < compare values. That's it. No mystery.
The > symbol means "greater than." The open side always points to the larger number.
The < symbol means "less than." The pointed side always points to the smaller number.
Here's the trick that actually sticks: think of the symbols as hungry mouths. They always want to eat the bigger number.
5 > 3 means 5 is greater than 3. The mouth opens toward 5 because 5 is bigger.
2 < 8 means 2 is less than 8. The mouth opens toward 8 because 8 is bigger.
The equal sign = means both sides match exactly.
Comparing Negative Numbers
This trips people up constantly. Here's the rule: on a number line, numbers to the right are always bigger. Negative numbers work the same way.
-3 > -7 because -3 sits to the right of -7 on the number line. -3 is closer to zero, which makes it larger.
-5 < -1 because -5 is further left than -1.
Think of it this way: -10 degrees is colder than -2 degrees. The temperature that's less negative is actually warmer.
Comparing Fractions and Decimals
Convert everything to the same format first. Fractions become decimals, or decimals become fractions. Pick whichever is easier.
Compare 1/4 and 0.3:
0.25 vs 0.30. Now it's obvious—0.3 is bigger.
Compare 3/5 and 2/3:
0.6 vs 0.666... The 2/3 wins.
Order of Operations: The Real Problem
Most people get this wrong. They solve left to right without thinking, and end up with answers that don't match reality.
The order of operations tells you which calculations come first. Without it, math becomes chaos where everyone gets different answers to the same problem.
The PEMDAS Rule
Parentheses → Exponents → Multiplication and Division → Addition and Subtraction
Parentheses first. Always. Whatever's inside them gets calculated first, no matter what.
Exponents come second. Powers and roots.
Multiplication and division are tied for third. You do them left to right, whichever appears first.
Addition and subtraction come last. Again, left to right.
The BODMAS Alternative
Same thing, different name. British and some other countries use BODMAS:
Brackets → Orders (exponents) → Division and Multiplication → Addition and Subtraction
Parentheses = Brackets. Same concept. Don't let the different terminology confuse you.
Why Multiplication and Division Aren't Separate Steps
Students memorize "PEMDAS" and then make the same mistake: they do all multiplication before any division. That's wrong.
8 ÷ 4 × 2
If you do multiplication first: 8 ÷ (4 × 2) = 8 ÷ 8 = 1
Correct answer: (8 ÷ 4) × 2 = 2 × 2 = 4
Work left to right when operators have the same priority. This applies to addition and subtraction too.
Why Addition and Subtraction Are Tied
10 - 3 + 2
Wrong approach: do addition first = 10 - 5 = 5
Correct approach: left to right = 7 + 2 = 9
Same thing with subtraction after addition. Work from left to right when the operators match.
Working Through Examples
Example 1: Basic PEMDAS
Solve: 3 + 6 × 2
Wrong answer: (3 + 6) × 2 = 18
Correct answer: 3 + (6 × 2) = 3 + 12 = 15
Multiplication comes before addition. No parentheses, so skip that step. Multiply first, then add.
Example 2: With Parentheses
Solve: (4 + 2) × 3 - 8 ÷ 2
Step 1: Parentheses → (4 + 2) = 6
Now we have: 6 × 3 - 8 ÷ 2
Step 2: Multiplication and division left to right
6 × 3 = 18
18 - 8 ÷ 2
8 ÷ 2 = 4
Step 3: Subtraction
18 - 4 = 14
Example 3: Nested Parentheses
Solve: 2 × (3 + (4 - 1) × 2)
Work inside out. Start with the innermost parentheses.
4 - 1 = 3
Now: 2 × (3 + 3 × 2)
Still have parentheses. Next: 3 × 2 = 6
Now: 2 × (3 + 6)
3 + 6 = 9
2 × 9 = 18
Example 4: Exponents Included
Solve: 2³ + 4 × (6 - 2)² ÷ 2
Parentheses first: 6 - 2 = 4
Now: 2³ + 4 × 4² ÷ 2
Exponents next: 2³ = 8 and 4² = 16
Now: 8 + 4 × 16 ÷ 2
Multiplication and division left to right:
4 × 16 = 64
64 ÷ 2 = 32
Now: 8 + 32 = 40
Common Mistakes That Kill Your Answer
- Ignoring parentheses entirely
- Doing multiplication before division (or vice versa) instead of left to right
- Adding before subtracting instead of working left to right
- Forgetting that subtraction can change results when parentheses are involved
- Mixing up the order of negative signs with the order of operations
Symbol Reference Table
| Symbol | Meaning | Example | Result |
|---|---|---|---|
| > | Greater than | 7 > 5 | True |
| < | Less than | 3 < 9 | True |
| = | Equal to | 6 = 6 | True |
| ≥ | Greater than or equal | 5 ≥ 5 | True |
| ≤ | Less than or equal | 4 ≤ 4 | True |
| ≠ | Not equal to | 3 ≠ 4 | True |
Operator Priority Table
| Priority | Operation | Example |
|---|---|---|
| 1st | Parentheses/Brackets | (2 + 3) |
| 2nd | Exponents/Orders | 5², √9 |
| 3rd | Multiply ÷ Divide | Left to right |
| 4th | Add ÷ Subtract | Left to right |
How to Approach Any Math Problem
Follow this sequence every time, without exception:
- Read the entire problem first. Don't start calculating until you see all the numbers and symbols.
- Identify parentheses. Circle or highlight them. These groups calculate first.
- Find exponents. Powers and roots come next.
- Work through multiplication and division. Left to right. Write down intermediate results.
- Finish with addition and subtraction. Left to right. Same process.
- Check your work. Run through the steps again. One missed parenthesis ruins everything.
The Bottom Line
Greater than and less than symbols compare values. The open side points to the bigger number. That's the whole concept.
Order of operations exists so everyone solves problems the same way. Parentheses first, then exponents, then multiplication/division left to right, then addition/subtraction left to right. Memorize it. Practice it. Check your work.
These aren't complicated ideas. The confusion comes from rushing. Slow down, follow the steps, and you'll get the right answer every time.