Compound Inequalities

A compound inequality joins two inequalities with the word and or the word or. That single word decides everything about the answer.

• “and” means intersection: the solution is the set of values that satisfy both inequalities. Graphically, it is the overlap of the two shaded regions.
• “or” means union: the solution is the set of values that satisfy at least one. Graphically, it is everything either region covers.

Each individual inequality is solved with the method from Linear Inequalities. The compound part is just deciding how to combine the two solution sets.

A quick way to keep the words straight: an and compound is stricter — a value has to clear two hurdles, so the solution set is usually smaller than either piece alone, and often a single bounded interval. An or compound is looser — clearing one hurdle is enough, so the solution set is usually larger than either piece, and often two disconnected rays.

The Compound Inequality Formula

The three-part form on the left is a compact “and”. The “or” form on the right stays as two separate pieces.

Solving an “and” Compound (Three-Part Form)

Keep the variable in the middle and apply every operation to all three parts.

Add to every part:

Divide every part by :

Interval: .

Solving an “and” Compound Written as Two Inequalities

Solve each separately, then take the overlap.

Interval: . If the two pieces did not overlap, the answer would be .

Solving an “or” Compound

Solve each separately, then take the union — keep both pieces.

Interval: . If the two pieces overlapped or touched to cover the line, the answer would be .

Graphing: Intersection vs. Union

Graph both pieces on the same number line first, then read off the result:

• “and” — the solution is where the two shadings overlap. If they never meet, there is no solution. The three-part form always graphs as one connected segment.
• “or” — the solution is everything either shading covers. If the two shadings together leave no gap, the solution is the whole line.

A useful check: an “and” answer written as an interval should have the smaller number on the left and be a single connected piece; an “or” answer should be two intervals joined by , or the whole line.

Worked Example A: Three-Part “and”

Solve and write the answer in interval notation.

Step 1 — subtract from all three parts:

Step 2 — divide all three parts by :

Answer: . Open circle at , filled circle at , shaded between.

Worked Example B: “and” With No Overlap

Solve .

Solve each:

Overlap: there is no number that is both less than and greater than .

Answer: no solution, .

Worked Example C: “or” With Two Pieces

Solve .

Solve each:

Union: keep both pieces.

Answer: .

Worked Example D: Dividing the Three-Part Form by a Negative

Solve .

Step 1 — subtract from all three parts:

Step 2 — divide all three parts by — both symbols flip:

Step 3 — rewrite with the smaller number on the left:

Answer: .

Worked Example E: An “or” That Covers Everything

Solve .

Solve each:

Union: graph (a ray going left from ) and (a ray going right from ). Together they cover every point on the line — anything less than or greater than is every real number.

Answer: .

Worked Example F: A Word Condition

A recipe works only when the oven temperature is at least and no more than . Write the acceptable range and its interval.

Both conditions must hold at once, so this is an “and” compound, and it fits the three-part form directly:

Answer: . “At least” gives (bracket) and “no more than” gives (bracket), so both endpoints are included.

Common Mistakes to Avoid

• Operating on only two of the three parts. In the three-part form, every step touches all three parts.
• Turning an “or” into a single interval. or stays as two pieces joined by ; it is not (that would be the region between, which is exactly what the “or” excludes).
• Forgetting to flip in the three-part form. Dividing all three parts by a negative flips both symbols; then reorder so the smaller value is on the left.
• Reporting an “and” with no overlap as an interval. No overlap means .
• Missing that an “or” can be all reals. If the pieces cover everything, the answer is .
• Confusing “at least” / “at most” wording. “At least ” is ; “at most ” is ; “between and ” is usually unless “inclusive” is stated.

Where This Shows Up Later

• Absolute Value Inequalities. is exactly the “and” compound ; is the “or” compound or .
• Domain of a function. A domain like ” and ” is a compound condition.
• Polynomial and Rational Inequalities. The solution is often a union of intervals, read the same way as an “or” result.
• Systems of linear inequalities. In two variables, the feasible region is the intersection (“and”) of several half-planes.

Practice Problems

Work each problem yourself before opening the answer. Write solutions in interval notation.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve .

Problem 13. Solve .

Problem 14. Solve .

Quick Reference

Each half is a Linear Inequality; the answer format is Interval Notation. Compound inequalities are the engine behind Absolute Value Inequalities. More Algebra lessons are available too.