Essential Rules for Solving Inequalities
What Are Inequalities and Why You Need to Master Them
Inequalities are mathematical statements that compare two expressions using symbols like <, >, ≤, or ≥. Unlike equations, which have a single solution, inequalities usually have a range of solutions.
You encounter inequalities in:
- Calculus when finding domains and ranges
- Optimization problems in business and economics
- Real-world scenarios like budget constraints
- Computer science for defining conditions
Most students struggle with inequalities because they apply equation rules incorrectly. One critical difference trips everyone up: flipping the inequality sign when multiplying or dividing by negative numbers.
The Core Rules for Solving Inequalities
These rules work exactly like equation rules, except when a negative number enters the picture.
Rule 1: Addition and Subtraction
You can add or subtract any number from both sides without changing the inequality's direction.
Example:
x - 3 < 7
x - 3 + 3 < 7 + 3
x < 10 ✓
Rule 2: Multiplication and Division by Positive Numbers
Multiplying or dividing both sides by a positive number keeps the inequality sign pointing the same direction.
Example:
2x ≤ 14
2x ÷ 2 ≤ 14 ÷ 2
x ≤ 7 ✓
Rule 3: Multiplication and Division by Negative Numbers
This is where most people fail. When you multiply or divide by a negative number, you must flip the inequality sign.
Example:
-4x > 12
-4x ÷ -4 < 12 ÷ -4 (sign flips!)
x < -3 ✓
Rule 4: Combining Inequalities
You can add inequalities with the same direction:
If a < b and c < d, then a + c < b + d
You cannot subtract inequalities reliably. Stick to addition.
How to Solve Inequalities: Step-by-Step
Here's the process that works every time:
- Simplify both sides — combine like terms, distribute
- Move variables to one side — use addition/subtraction
- Isolate the variable — multiply/divide, watching for negatives
- Check your work — plug a test value into the original inequality
Worked Example:
Solve: 3(x - 2) + 5 ≥ 2x + 1
Step 1: Distribute
3x - 6 + 5 ≥ 2x + 1
Step 2: Combine like terms
3x - 1 ≥ 2x + 1
Step 3: Subtract 2x from both sides
x - 1 ≥ 1
Step 4: Add 1 to both sides
x ≥ 2
Check: Test x = 3
3(3 - 2) + 5 ≥ 2(3) + 1
3 + 5 ≥ 6 + 1
8 ≥ 7 ✓
Solution: x ≥ 2
Compound Inequalities
Compound inequalities combine two inequalities into one statement. There are two types.
Intersection (And)
Both conditions must be true simultaneously.
Example: -2 < x + 3 ≤ 5
Break it into two parts:
-2 < x + 3 and x + 3 ≤ 5
-5 < x and x ≤ 2
Solution: -5 < x ≤ 2
Union (Or)
Either condition being true satisfies the inequality.
Example: x < -1 or x > 3
Solution includes all x less than -1 or all x greater than 3.
Absolute Value Inequalities
These require splitting into two cases.
Rule: |x| < a becomes -a < x < a
Rule: |x| > a becomes x < -a or x > a
Example: |2x - 1| < 5
-5 < 2x - 1 < 5
Add 1: -4 < 2x < 6
Divide by 2: -2 < x < 3
Common Mistakes to Avoid
| Mistake | What Actually Happens | Fix |
|---|---|---|
| Forgetting to flip the sign | Multiplying by negative without changing direction | Always check: did a negative multiply/divide occur? |
| Reversing compound inequalities | Writing -3 > x > 5 instead of 5 < x < -3 | Always rewrite with smaller number first |
| Multiplying by variable expressions | Unknown sign changes inequality direction unpredictably | Never multiply by variables unless you know the sign |
| Dropping absolute value bars | Only capturing one case | Always split into positive and negative cases |
Graphing Inequalities on a Number Line
Visualizing inequalities helps you understand the solution set.
- Open circle (< or >) — the endpoint is NOT included
- Closed circle (≤ or ≥) — the endpoint IS included
- Shade toward the direction the inequality points
Example: x > -1 and x ≤ 3
Draw an open circle at -1, shade right. Draw a closed circle at 3, shade left. The overlap (where both conditions apply) is your solution: -1 < x ≤ 3
Quick Reference: Inequality Rules Summary
| Operation | Positive Number | Negative Number |
|---|---|---|
| Add/Subtract | Sign stays same | Sign stays same |
| Multiply/Divide | Sign stays same | Sign flips |
That's the entire rule set. Everything else is just applying these consistently.
Getting Started: Practice Problems
Try these three, then check your answers.
1. Solve: 5 - 2x < 17
2. Solve: -3 ≤ 2x + 1 < 7
3. Solve: |x + 4| ≥ 3
Answers:
1. 5 - 2x < 17
-2x < 12
x > -6 (flipped because dividing by -2)
2. -3 ≤ 2x + 1 < 7
-4 ≤ 2x < 6
-2 ≤ x < 3
3. |x + 4| ≥ 3 becomes x + 4 ≤ -3 or x + 4 ≥ 3
x ≤ -7 or x ≥ -1
Final Warning
The only thing that separates correct inequality solutions from wrong ones is attention to sign changes. Every time you multiply or divide by a negative, stop and explicitly flip the sign. Don't assume you'll remember. Write it down.
Master this, and every inequality problem becomes straightforward.