MathematicsChapter 4: Linear and Quadratic Inequalities
Compound Inequalities
A compound inequality joins two conditions with and (intersection, overlap) or or (union, either piece).
Learning objectives
- Explain compound inequalities using the correct definitions, conditions, and notation.
- Apply and = overlap, or = union correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
A compound inequality joins two conditions with and (intersection, overlap) or or (union, either piece). 'And' is a sandwich on the number line: -2 < x <= 5. 'Or' is two separate rays. Solving 1 < 2x+3 < 9 means both ends at once: subtract 3, then divide by 2.
Where / symbols
- COMPOUND INEQUALITIES
- A compound inequality joins two conditions with and (intersection, overlap) or or (union, either piece). 'And' is a sandwich on the number line: -2 < x <= 5. 'Or' is two separate rays. Solving 1 < 2x+3 < 9 means both ends at once: subtract 3, then divide by 2.
- AND (SANDWICH)
- Keep the inequality directions when you add/subtract the same number from all three parts. Divide by a positive; if you divide by a negative, reverse both signs. Graph: filled/open at both ends, bar between.
- OR
- Solve each piece, then unite. x<-2 or x>=5. Do not turn that into -2<x<5. That would be the opposite (the gap).
Relationship
- and = overlap
- or = union
Overview
One line or two rays:
And (sandwich)
Keep the inequality directions when you add/subtract the same number from all three parts. Divide by a positive; if you divide by a negative, reverse both signs. Graph: filled/open at both ends, bar between.
Examples 1 < 2x+3 < 9 · subtract 3: -2 < 2x < 6 · divide 2: -1 < x < 3
Or
Solve each piece, then unite. x<-2 or x>=5. Do not turn that into -2<x<5. That would be the opposite (the gap).
Examples x<-2 or x>=5 · Two rays · Not the segment between
WHAT — COMPOUND INEQUALITIES
A compound inequality joins two conditions with and (intersection, overlap) or or (union, either piece). 'And' is a sandwich on the number line: -2 < x <= 5. 'Or' is two separate rays. Solving 1 < 2x+3 < 9 means both ends at once: subtract 3, then divide by 2. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Compound Inequalities matters because inequalities describe ranges and limits rather than one exact value. They are used for constraints, graphs, optimisation, and real situations such as budgets, dimensions, and capacity.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through AND (SANDWICH), then OR. Worked: solve 1 < 2x + 3 < 9. 1 < 2x + 3 < 9 subtract 3 from all three parts -2 < 2x < 6 divide all three parts by 2 (positive, no flip) -1 < x < 3 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- solve 1 < 2x + 3 < 9.
- 1 < 2x + 3 < 9
- subtract 3 from all three parts
- -2 < 2x < 6
- divide all three parts by 2 (positive, no flip)
- -1 < x < 3
In simple words
Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
Key points
- and = overlap
- or = union
- all three parts
- unite the two solutions
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Compound Inequalities
Sticky Notes · Quick Revision
Core Idea
- A compound inequality joins two conditions with and (intersection, overlap) or or (union, either piece). 'And' is a sandwich on the number line: -2 < x <= 5. 'Or' is two separate rays. Solving 1 < 2x+3 < 9 means both ends at once: subtract 3, then divide by 2.
AND (SANDWICH)
- Keep the inequality directions when you add/subtract the same number from all three parts. Divide by a positive; if you divide by a negative, reverse both signs. Graph: filled/open at both ends, bar between.
OR
- Solve each piece, then unite. x<-2 or x>=5. Do not turn that into -2<x<5. That would be the opposite (the gap).
Worked Method
- Worked: solve 1 < 2x + 3 < 9.
- 1 < 2x + 3 < 9
- subtract 3 from all three parts
- -2 < 2x < 6
- divide all three parts by 2 (positive, no flip)
- -1 < x < 3
Common Misconception
- Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
FBISE Exam Check
- Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
60-second revision
- A compound inequality joins two conditions with and (intersection, overlap) or or (union, either piece).
- AND (SANDWICH): Keep the inequality directions when you add/subtract the same number from all three parts.
- OR: Solve each piece, then unite.
- Worked-method reminder: -1 < x < 3
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