Compound Inequalities Worksheet- Practice and Solutions
What Are Compound Inequalities?
Compound inequalities are two simple inequalities joined together by either "and" or "or." That's it. Nothing fancy. You solve each part separately, then figure out what values satisfy both conditions (for "and") or at least one condition (for "or").
Most students either nail these or completely bomb them. There's not much middle ground. The reason is simple: you either understand the logic or you don't. Practice worksheets fix that gap faster than anything else.
The Two Types You Need to Know
AND Inequalities (Intersection)
When you see "and," the solution must satisfy both inequalities. Think of it like a Venn diagram where only the overlap counts.
Example: 2 < x ≤ 5 means x is greater than 2 AND less than or equal to 5.
OR Inequalities (Union)
When you see "or," the solution satisfies at least one of the inequalities. If either condition is true, the value works.
Example: x < -1 or x > 3 means x can be less than -1, or x can be greater than 3. Both ranges count.
How to Solve Compound Inequalities
Here's the process. No shortcuts, no tricks.
- Break the compound inequality into its two parts
- Solve each inequality independently
- For "and": find the overlap where both solutions meet
- For "or": combine both solution sets
- Write the final answer in interval notation or inequality notation
That's the whole method. Practice makes it automatic.
Practice Problems with Solutions
Work through these. Check your answers only after you've tried.
Problem 1: Solve -3 ≤ 2x - 1 < 5
This is an AND inequality written as a compound statement.
Step 1: Break it apart
- -3 ≤ 2x - 1
- 2x - 1 < 5
Step 2: Solve each
- Add 1: -2 ≤ 2x
- Divide by 2: -1 ≤ x
- Add 1: 2x < 6
- Divide by 2: x < 3
Answer: -1 ≤ x < 3 or [-1, 3) in interval notation
Problem 2: Solve x + 4 < 2 or 3x > 12
This is an OR inequality.
Step 1: Solve each inequality
- x + 4 < 2 → x < -2
- 3x > 12 → x > 4
Step 2: Combine (it's OR, so both ranges count)
Answer: x < -2 or x > 4, or (-∞, -2) ∪ (4, ∞) in interval notation
Problem 3: Solve -5 < 3 - 2x ≤ 1
Another AND compound inequality.
Step 1: Break it apart
- -5 < 3 - 2x
- 3 - 2x ≤ 1
Step 2: Solve each
- Subtract 3: -8 < -2x
- Divide by -2 (flip sign): 4 > x
- Subtract 3: -2x ≤ -2
- Divide by -2 (flip sign): x ≥ 1
Answer: 1 ≤ x < 4 or [1, 4) in interval notation
Common Mistakes That Cost You Points
- Forgetting to flip the sign when dividing by a negative number. This single error ruins everything that follows.
- Mixing up AND and OR. AND means intersection. OR means union. Different answers.
- Graphing errors. Use open circles for strict inequalities (< or >) and closed circles for inclusive ones (≤ or ≥).
- Writing interval notation wrong. Parentheses mean "not included." Brackets mean "included." A parenthesis next to a bracket is fine.
Where to Get Practice Worksheets
You need problems. Lots of them. Here's how the main options stack up.
| Resource | Pros | Cons |
|---|---|---|
| Khan Academy | Free, immediate feedback, video explanations | Limited worksheet format, requires internet |
| Kuta Software | Generates unlimited problems, answer keys included | Requires purchase for full access |
| Math-Aids.com | Free customizable worksheets, variety of formats | Answers sometimes have errors |
| School textbooks | Curated problems, matches class curriculum | Often too few problems per topic |
| Chegg Study | Step-by-step solutions, large problem bank | Subscription required, expensive |
For most people, Kuta Software or Math-Aids.com gives you the best practice-to-effort ratio. Khan Academy works if you need the video explanations first.
Getting Started: Your Practice Routine
Don't just read through problems. That does nothing.
- Start with 5 problems per day. Focus on one type (AND or OR) until you get 4 out of 5 right.
- Time yourself. You should solve a standard compound inequality in under 2 minutes once you're competent.
- Check answers immediately. Wrong habits calcify fast if you practice mistakes.
- Graph the solutions. Number line graphing reinforces the logic better than just writing interval notation.
- Mix types randomly. Tests won't tell you which type is coming. Get comfortable switching between AND and OR.
Do this for a week. You'll see the difference.
Quick Reference Cheat Sheet
- AND: Solution is the overlap. Write with a single variable expression when possible (like -3 < x ≤ 7).
- OR: Solution is the combined range. Write both conditions (like x < 2 or x > 5).
- Interval notation: Parentheses for strict, brackets for inclusive. Use ∪ for OR unions.
- Number lines: Open circle = not included. Closed circle = included.
Keep this handy while you practice. Refer back when you're stuck.