What is an Inequality?
An inequality compares two mathematical values using inequality signs instead of an equal sign:
- •<: Less than (open circle on number line)
- •>: Greater than (open circle on number line)
- •≤: Less than or equal to (closed / solid circle)
- •≥: Greater than or equal to (closed / solid circle)
Unlike equations that typically have one exact answer (like x = 4), inequalities represent a whole range of numbers (like x > 4, which includes 4.1, 5, 100, \dots).
The Golden Rule: When to Flip the Sign
The most famous rule in algebra is:
Whenever you MULTIPLY or DIVIDE both sides by a NEGATIVE number, you MUST FLIP the inequality sign!
* < flips to >
* > flips to <
* ≤ flips to ≥
* ≥ flips to ≤
Why does this happen? Intuitive Proof:
Consider a true statement:
Now multiply both numbers by -1:
- •Left side becomes -2
- •Right side becomes -5
Which is bigger: -2 or -5?
On a thermometer or number line, -2$ is warmer (greater) than $-5!
Therefore:
The sign had to flip to keep the statement mathematically true!
Worked Example 1: Standard Two-Step Inequality
Problem:
Solve and describe the number line graph:
3x - 7 > 8
Step 1: Add 7 to both sides
(Note: Adding never flips the sign!)
Step 2: Divide both sides by 3
(Note: 3 is positive, so the sign stays as >!)
How to Graph x > 5 on a Number Line:
- Put an open circle at 5 (because 5 is not included).
- Shade an arrow pointing to the right towards larger numbers (6, 7, 8, \dots).
Worked Example 2: Multi-Step with Sign Flip
Problem:
Solve the multi-step inequality:
-5x + 12 ≥ 37
Step 1: Subtract 12 from both sides
Step 2: Divide both sides by -5
Here comes the golden rule! Because we are dividing by -5, the ≥ symbol flips to ≤:
Final Answer: x ≤ -5.
How to Graph x ≤ -5:
- Place a solid, closed circle at -5 (because of the "or equal to" bar).
- Shade an arrow pointing to the left towards smaller numbers (-6, -7, \dots).
Number Line Quick Guide
| Symbol | Meaning | Circle Type on Graph | Arrow Direction |
|---|---|---|---|
| x > c | Greater than | Open Circle (\circ) | Points Right ( → ) |
| x < c | Less than | Open Circle (\circ) | Points Left (\leftarrow) |
| x ≥ c | Greater than or equal | Closed Solid (\bullet) | Points Right ( → ) |
| x ≤ c | Less than or equal | Closed Solid (\bullet) | Points Left (\leftarrow) |
When NOT to Flip the Sign!
Many students flip the sign whenever they see ANY minus sign in a problem.
If you solve 4x = -20 ⟹ x = -5, the sign does NOT* flip because the number dividing (4) is positive!
* Only flip when the number attached to x that you are dividing or multiplying by is negative.