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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.
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Compound Inequalities
What is it?
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.
In simple words
- 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
Important terms
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).
What you should learn
- 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.
Examples
- 1 < 2x+3 < 9
- subtract 3: -2 < 2x < 6
- divide 2: -1 < x < 3
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).
Recommended for Class 10
Full school-level explanation
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.
One line or two rays:
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.
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.
Remember
- all three parts
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).
Remember
- unite the two solutions
Examples
- x<-2 or x>=5
- Two rays
- Not the segment between
Remember
- and = overlap
- or = union
- -1 < x < 3
- and, not or
- all three parts
- unite the two solutions
- and = overlap
- or = union
How the ideas connect
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
Deeper understanding
Go beyond memorising Compound Inequalities. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
AND (SANDWICH)
OR
Worked Method
Exam focus
Quick revision
- and = overlap
- or = union
- -1 < x < 3
- and, not or
- COMPOUND INEQUALITIES: A compound inequality joins two conditions with and (intersection, overlap) or or (union, either pie
- AND (SANDWICH): Keep the inequality directions when you add/subtract the same number from all three parts. Divide by
Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
For long answers on Compound Inequalities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- and = overlap
- or = union
- -1 < x < 3
- and, not or
Important formulas
- and = overlap
- or = union
- -1 < x < 3
Common exam traps
- Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
- Not the segment between
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
- Mistake
Not the segment between
- Mistake
Do not treat AND (SANDWICH) and OR as the same step; each has a different purpose.
- Mistake
Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
and = overlap
or = union
-1 < x < 3
and, not or
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. 'O
and = overlap
- 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. 'O
- Formula
- and = overlap
- Example
- 1 < 2x+3 < 9
- Critical warning
- Do not split a sandwich into two equations connected by 'or'. 1<2x+3 and 2x+3<9 must both hold.
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Exam Practice
Define
Q1. Define Compound Inequalities.
Explain
Q2. Explain: AND (SANDWICH).
Example
Q3. Give one clear example of Compound Inequalities and say why it fits.
Mistake
Q4. State one common mistake students make about Compound Inequalities.
Exam note
Q5. Write a short exam note on Compound Inequalities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.