Compound Inequalities

Learn to solve and graph compound inequalities using AND and OR connectors.

Advanced25 minLesson
Printable worksheet

Definition

A compound inequality combines two inequalities using the words AND or OR.
AND Inequalities (Conjunction):
  • Both conditions must be true at the same time
  • Written as: a<x<ba < x < b or "x>ax > a AND x<bx < b"
  • The solution is the intersection (overlap) of both inequalities
OR Inequalities (Disjunction):
  • At least one condition must be true
  • Written as: "x<ax < a OR x>bx > b"
  • The solution is the union (combination) of both inequalities
AND=∩ (intersection)OR=∪ (union)\text{AND} = \cap \text{ (intersection)} \qquad \text{OR} = \cup \text{ (union)}

Try it now

Which compound inequality represents "xx is greater than −3-3 AND less than 55"?

Worked Examples

Solve: −3<2x+1<7-3 < 2x + 1 < 7
1

Understand the compound inequality

This means −3<2x+1-3 < 2x + 1 AND 2x+1<72x + 1 < 7 simultaneously → Both must be true
2

Subtract 1 from all three parts

−3−1<2x+1−1<7−1-3 - 1 < 2x + 1 - 1 < 7 - 1 → −4<2x<6-4 < 2x < 6
3

Divide all three parts by 2

−42<2x2<62\frac{-4}{2} < \frac{2x}{2} < \frac{6}{2} → −2<x<3-2 < x < 3
4

Write in interval notation

Open circles at −2-2 and 33 (not included) → (−2,3)(-2, 3)

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 5<x<25 < x < 2 and thinking it has solutions

Why it's wrong: This would require xx to be greater than 5 AND less than 2 simultaneously, which is impossible.

Correct: Always check if an AND inequality makes logical sense. If a>ba > b in a<x<ba < x < b, there is no solution.

Using the wrong interval notation brackets

Why it's wrong: Square brackets [[ ]] mean the endpoint is included (≤\leq or ≥\geq), parentheses (( )) mean it's excluded (<< or >>).

Correct: x>3x > 3: use (3,∞)(3, \infty). x≥3x \geq 3: use [3,∞)[3, \infty). Infinity always uses parentheses.

Watch a video explanation

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How to Solve Compound Inequalities | Step-by-Step Examples and Graphs

by Understand The Math

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Interactive Visual

to

-2 < x < 3·(-2, 3)

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to

x < -2 or x > 2·(−∞, -2) ∪ (2, ∞)

Click on numbers to select them. Adjust the range to explore different values.

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Practice Problems

16 problems
Problem 1 of 16
Easy
Which compound inequality represents "xx is greater than −3-3 AND less than 55"?

Why It Matters

Compound inequalities describe real-world situations where multiple conditions apply:
  • Temperature ranges: A comfortable room is between 18 and 24 degrees Celsius (18≤T≤2418 \leq T \leq 24)
  • Speed limits: You must drive at least 45 km/h but no more than 120 km/h on a highway (45≤s≤12045 \leq s \leq 120)
  • Age restrictions: A movie is for viewers under 12 OR over 18 (a<12a < 12 OR a>18a > 18)
  • Test scores: You pass if you score at least 60 AND complete all assignments
Understanding compound inequalities helps you analyze situations with multiple constraints!

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: 114≤h≤133114 \leq h \leq 133

1Try It Yourself

Your maximum heart rate is 200 bpm. The cardio zone is 70% to 85% of maximum.

Write the compound inequality for heart rate hh in the cardio zone.

Step 1: Choose the right answer

Calculate 70% and 85% of 200. Which compound inequality fits?

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: 9.8≤d≤10.29.8 \leq d \leq 10.2

2Try It Yourself

A factory rejects items weighing less than 95 grams OR more than 105 grams.

Write the inequality for rejected items (weight ww).

Step 1: Choose the right answer

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: 60≤s≤13060 \leq s \leq 130

3Try It Yourself

A ticket is issued if you drive under 50 km/h OR over 110 km/h on a specific road.

For what speeds ss would you get a ticket?

Step 1: Choose the right answer

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

AND means both conditions must be true simultaneously (intersection). OR means at least one condition must be true (union). AND typically gives a smaller solution set; OR gives a larger one.
AND means both conditions must be true simultaneously (intersection). OR means at least one condition must be true (union). AND typically gives a smaller solution set; OR gives a larger one.
Use square brackets [][ ] when the endpoint is included (≤\leq or ≥\geq). Use parentheses ()( ) when the endpoint is NOT included (<< or >>). Infinity (∞\infty) ALWAYS uses parentheses because you can never reach infinity.
Yes! An AND inequality has no solution when the conditions contradict each other (like x>5x > 5 AND x<2x < 2). However, an OR inequality will always have a solution unless the individual inequalities themselves have no solutions.

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: (a,b)(a, b), [a,b][a, b], [a,b)[a, b), (a,b](a, b]

Formula Card

AND Inequality (Conjunction)

a<x<ba < x < b means x>ax > a AND x<bx < b

Solution is the intersection - both conditions must be true

OR Inequality (Disjunction)

x<ax < a OR x>bx > b

Solution is the union - at least one condition must be true

Interval Notation

(a,b)=a<x<b(a, b) = a < x < b, [a,b]=a≤x≤b[a, b] = a \leq x \leq b

Parentheses = excluded, Brackets = included, infinity always uses parentheses

Today's challengeOne problem, three tries

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