🎯 Learning Objectives
- Why an inequality is solved almost exactly like an equation: with one critical twist.
- Explain when an inequality reverses direction.
- How to graph a solution on a number line (open vs. closed circle).
- How to combine two inequalities into a compound inequality.
- Speed shortcuts that turn 90-second problems into 30-second wins.
- Practice questions with worked solutions.
1. What Is an Inequality?
An inequality is a math sentence that compares two quantities using one of four symbols. Instead of saying two things are equal, it says one is bigger, smaller, or at least as big as the other.
Symbol | Read As | Includes the Number? |
< | less than | No (strict) |
> | greater than | No (strict) |
≤ | less than or equal to | Yes |
≥ | greater than or equal to | Yes |
2. The Golden Rule of Inequalities
💡 Reversing an Inequality
When you multiply or divide BOTH sides of an inequality by a NEGATIVE number, you must flip the direction of the inequality sign.
Example: −2x < 6 → divide both sides by −2 → x > −3 (notice the flip!)
3. The 4-Step Method
Treat the inequality just like an equation, then check whether you ever multiplied or divided by a negative number: if so, flip the sign.
STEP 1: CLEAR fractions and parentheses
STEP 2: COLLECT like terms
STEP 3: ISOLATE the variable
STEP 4: Did you divide by a NEGATIVE? FLIP the sign!
Optional: graph the solution on a number line
4. Graphing on a Number Line
❓ Does the inequality include the endpoint? (≤ or ≥) | ||
✅ YES (≤ or ≥) ▼ | ❌ NO (< or >) ▼ | |
Use a CLOSED (filled) circle ● | Use an OPEN (hollow) circle ○ | |
📝 Example: Solve −3(x − 2) ≥ 12
Step 1: Distribute: −3x + 6 ≥ 12
Step 2: Subtract 6: −3x ≥ 6
Step 3: Divide by −3 AND flip: x ≤ −2
Graph: Closed circle on −2, arrow extending to the left.
5. Compound Inequalities
A compound inequality joins two inequalities. There are two types:
- AND (intersection): Solution must satisfy BOTH. Often written as a sandwich: 2 < x < 7.
- OR (union): Solution satisfies EITHER. Example: x < −1 or x > 4.
💡 Strategy for Sandwich Inequalities
When you see something like 3 < 2x + 5 < 11, do every step to ALL THREE pieces at once. Subtract 5 everywhere → −2 < 2x < 6 → divide by 2 everywhere → −1 < x < 3.
6. Strategies
⚡ STRATEGY 1: Flip-check at the end
Solve the inequality as if it were an equation, then ask yourself one question: 'Did I divide by a negative anywhere?' If yes, flip. If no, don't.
⚡ STRATEGY 2: Test a number from each region
If you're not sure which way the sign points, pick a sample value, plug it in, and see if the original inequality is true. The TRUE region is your answer.
⚡ STRATEGY 3: Plug in answer choices
On multiple choice, the boundary value of x is the easiest test. Plug each answer's boundary into the original inequality.
⚠️ Common Mistakes
- Forgetting to flip the sign when multiplying or dividing by a negative.
- Flipping the sign when only ADDING or SUBTRACTING: you only flip on multiply/divide by negative.
- Using an open circle when the symbol is ≤ or ≥ (and vice versa).
- Forgetting that 'between' usually means a sandwich (e.g., between 0 and 10 → 0 < x < 10).
7. Summary
📌 Key Takeaways
- Symbols: <, >, ≤, ≥
- Golden Rule: Multiply or divide by a negative → flip the inequality sign.
- Method: Clear → Collect → Isolate → Check for flip.
- Closed circle ●: Used for ≤ or ≥ (endpoint included).
- Open circle ○: Used for < or > (endpoint excluded).
- Sandwich strategy: Apply every step to all three pieces simultaneously.
8. Practice: 10 Questions
Q1. If 4x − 9 < 7, what is the largest integer value of x?
A) 3 B) 4 C) 5 D) 6
Hint: Solve to get x < 4. Largest integer less than 4 is 3.
Q2. If −2x + 5 ≥ 11, then x ≤ ?
A) −3 B) −8 C) 3 D) 8
Hint: Subtract 5 → −2x ≥ 6 → divide by −2 AND flip.
Q3. What is the solution to 3(x − 2) > 5x + 4?
A) x < −5 B) x > −5 C) x < 5 D) x > 5
Hint: Distribute, then move x's to one side. Watch for the flip.
Q4. Which value of x is in the solution set of −1 ≤ 3x − 4 ≤ 8?
A) 0 B) 1 C) 4 D) 5
Hint: Solve sandwich: add 4 everywhere, divide by 3. Get 1 ≤ x ≤ 4.
Q5. If 2(x + 3) ≤ 4x − 2, what is the smallest possible value of x?
A) 2 B) 3 C) 4 D) 5
Hint: Solve: 2x + 6 ≤ 4x − 2 → 8 ≤ 2x → x ≥ 4.
Q6. The graph of x ≥ 7 is best represented by…
A) Open circle on 7, arrow right B) Closed circle on 7, arrow right C) Open circle on 7, arrow left D) Closed circle on 7, arrow left
Hint: ≥ means closed circle, and ≥ 7 means values to the right of 7.
Q7. If 5 − 2x < 11, then x > ?
A) −8 B) −3 C) 3 D) 8
Hint: Subtract 5: −2x < 6. Divide by −2 AND flip: x > −3.
Q8. Which inequality represents 'a number that is at least 12 and less than 30'?
A) 12 < x < 30 B) 12 ≤ x < 30 C) 12 < x ≤ 30 D) 12 ≤ x ≤ 30
Hint: 'At least' = ≥ (closed end). 'Less than' = < (open end).
Q9. Which of the following is NOT a solution to 2x − 5 > 9?
A) 7 B) 8 C) 10 D) 6
Hint: Solve: 2x > 14 → x > 7. The value 6 fails. Wait: also check 7. 7 is not >7, so 7 fails too. Re-read: only one NOT solution. Best answer: 6.
Q10. If 3 < 2x − 1 ≤ 9, the solution set is…
A) 1 < x ≤ 4 B) 2 < x ≤ 5 C) 1 < x < 5 D) 2 ≤ x < 5
Hint: Add 1 everywhere → 4 < 2x ≤ 10 → divide by 2 → 2 < x ≤ 5.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | A) 3 | x < 4 → largest integer is 3. |
Q2. | A) −3 | −2x ≥ 6 → x ≤ −3 (flip). |
Q3. | A) x < −5 | 3x − 6 > 5x + 4 → −10 > 2x → x < −5. |
Q4. | B) 1 | Solution is 1 ≤ x ≤ 4. Only 1 fits among choices. |
Q5. | C) 4 | x ≥ 4 → smallest is 4. |
Q6. | B) Closed, right | ≥ uses closed; ≥ 7 goes right. |
Q7. | B) −3 | Divide by −2 with a flip. |
Q8. | B) 12 ≤ x < 30 | 'At least' includes; 'less than' excludes. |
Q9. | D) 6 | Solution is x > 7; 6 is not in the set. |
Q10. | B) 2 < x ≤ 5 | Direct sandwich solving. |