Compound Inequalities
Learn to solve and graph compound inequalities using AND and OR connectors.
Definition
- Both conditions must be true at the same time
- Written as: or " AND "
- The solution is the intersection (overlap) of both inequalities
- At least one condition must be true
- Written as: " OR "
- The solution is the union (combination) of both inequalities
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Worked Examples
Solve:
Understand the compound inequality
This means AND simultaneously → Both must be true
Subtract 1 from all three parts
→
Divide all three parts by 2
→
Write in interval notation
Open circles at and (not included) →
Answer: or in interval notation:
Common Mistakes
Confusing AND with OR when graphing
Why it's wrong: AND requires the overlap (intersection), while OR includes everything from both sets (union).
Correct: AND = intersection (smaller region, both conditions). OR = union (larger region, either condition).
Forgetting to flip the inequality when multiplying/dividing by a negative
Why it's wrong: When solving each part of a compound inequality, the same rules apply as with regular inequalities.
Correct: If you multiply or divide by a negative number, reverse ALL inequality signs in that step.
Writing and thinking it has solutions
Why it's wrong: This would require to be greater than 5 AND less than 2 simultaneously, which is impossible.
Correct: Always check if an AND inequality makes logical sense. If in , there is no solution.
Using the wrong interval notation brackets
Why it's wrong: Square brackets mean the endpoint is included ( or ), parentheses mean it's excluded ( or ).
Correct: : use . : use . Infinity always uses parentheses.
Interactive Visual
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Practice Problems
16 problemsWhich compound inequality represents " is greater than AND less than "?
Why It Matters
- Temperature ranges: A comfortable room is between 18 and 24 degrees Celsius ()
- Speed limits: You must drive at least 45 km/h but no more than 120 km/h on a highway ()
- Age restrictions: A movie is for viewers under 12 OR over 18 ( OR )
- Test scores: You pass if you score at least 60 AND complete all assignments
Real World Applications
Healthy Heart Rate Zones
Fitness trainers use compound inequalities to define heart rate zones during exercise.
Example:
The fat-burning zone is when your heart rate is between 60% and 70% of your maximum. For a max of 190 bpm:
Your maximum heart rate is 200 bpm. The cardio zone is 70% to 85% of maximum.
Write the compound inequality for heart rate in the cardio zone.
Step 1: Write the mathematical expression
Calculate 70% of 200 and 85% of 200:
Manufacturing Quality Control
Products must meet specifications within tolerance ranges. Items outside the range are rejected.
Example:
A bolt must be between 9.8 mm and 10.2 mm to pass inspection:
A factory rejects items weighing less than 95 grams OR more than 105 grams.
Write the inequality for rejected items (weight ).
Step 1: Write the mathematical expression
Items are rejected if:
Safe Driving Speed
Traffic laws use compound inequalities to define legal speed ranges.
Example:
On a highway, you must drive at least 60 km/h but no more than 130 km/h:
A ticket is issued if you drive under 50 km/h OR over 110 km/h on a specific road.
For what speeds would you get a ticket?
Step 1: Write the mathematical expression
Ticket if speed is:
Key Takeaways
- 1Compound inequalities combine two inequalities using AND or OR
- 2AND inequalities require BOTH conditions to be true (intersection) - the solution is where the graphs overlap
- 3OR inequalities require AT LEAST ONE condition to be true (union) - the solution includes both regions
- 4When solving, treat each part separately, then combine the solutions
- 5AND can result in no solution if the conditions contradict; OR can include all real numbers if conditions overlap completely
Frequently Asked Questions
Glossary
- Compound inequality
- Two or more inequalities joined by AND or OR
- Conjunction (AND)
- A compound statement that is true only when BOTH parts are true
- Disjunction (OR)
- A compound statement that is true when AT LEAST ONE part is true
- Intersection
- The overlap of two sets; values that satisfy BOTH conditions
- Union
- The combination of two sets; values that satisfy EITHER condition
- Interval notation
- A way to write solution sets using brackets: , , ,
Formula Card
AND Inequality (Conjunction)
Solution is the intersection - both conditions must be true
OR Inequality (Disjunction)
Solution is the union - at least one condition must be true
Interval Notation
Parentheses = excluded, Brackets = included, infinity always uses parentheses