Linear Equations & Inequalities

Compound Inequalities

Solving and graphing 'and'/'or' compound inequalities.

Intermediate20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Solving Linear Inequalities

Combining Two Conditions at Once

Some situations need more than one boundary to describe correctly — a temperature that must stay between two values, or a coupon that only works above a minimum purchase but below a maximum. A compound inequality joins two inequalities with the word 'and' or the word 'or,' and each word changes what the combined solution looks like.

Definition — 'And' Compound Inequality

A statement true only when both inequalities are true at once, often written as a single chained inequality like 1 < x < 5. The solution is the overlap — every value that satisfies both conditions simultaneously.

Definition — 'Or' Compound Inequality

A statement true when at least one of the two inequalities is true. The solution is the combination of both individual solution sets, even where they don't overlap.

Worked Example — Solving an 'And' Inequality

Solve −3 < 2x + 1 < 9. Apply the same operation to all three parts at once. Subtract 1 from every part: −4 < 2x < 8. Divide every part by 2: −2 < x < 4. The solution is every number strictly between −2 and 4.

Worked Example — Solving an 'Or' Inequality

Solve x − 2 < −5 or x + 3 > 8. Solve each piece separately. First: x − 2 < −5 gives x < −3. Second: x + 3 > 8 gives x > 5. Since the two pieces are joined by 'or,' the solution is every number that satisfies at least one: x < −3 or x > 5.

Equation Editor

Edit as raw LaTeX

sum(i, 1, n, i^2) = n(n+1)(2n+1)/6

Constants

Structures

Calculus & discrete math

Greek

\sum_{i=1}^{n} i^{2} = \frac{\operatorname{n}\left(n + 1\right) \cdot \left(2 \cdot n + 1\right)}{6}

Evaluate

Unknown function "n"

Tip

Sketching each compound inequality on a number line makes the difference between 'and' and 'or' visual: 'and' solutions form one connected segment between two boundaries, while 'or' solutions form two separate rays pointing away from each other.

Common Mistakes

Applying an operation to only one part of a chained 'and' inequality, such as dividing only the middle of −4 < 2x < 8 by 2.
A chained inequality has three parts. Whatever operation is applied — adding, subtracting, multiplying, or dividing — must be applied to all three parts at the same time to keep the statement valid.
Combining an 'or' solution into one chained inequality, such as writing 5 < x < −3 for x < −3 or x > 5.
A chained inequality like a < x < b only makes sense when a is less than b and describes an 'and' relationship. Two disjoint 'or' solutions must stay written as two separate inequalities joined by the word 'or.'

Key Takeaways

An 'and' compound inequality's solution is the overlap between two conditions — often written as one chained inequality.
An 'or' compound inequality's solution combines both individual solution sets, even when they don't overlap.
Whatever operation you apply to an 'and' inequality must be applied to every part at once.

Summary

Compound inequalities let you describe precise ranges using 'and' and broader possibilities using 'or.' With equations and inequalities both in place, the next unit turns to graphing and writing the linear functions behind them.

Track your progress

← Solving Linear Inequalities Back to Linear Equations & Inequalities