MathematicsChapter 5: Linear Equations and Inequalities
Compound Inequalities
A compound inequality joins two statements with or or and.
Learning objectives
- Explain compound inequalities in clear mathematical language
- Apply the ideas in or means union
- Solve problems involving and means intersection
Definition
A compound inequality joins two statements with or or and. Or means union: a number is in the solution set if it works in at least one part. And means intersection: the number must work in both parts at once. A chain such as -2 < x < 5 is a short way to write x > -2 and x < 5, so a chain is AND, not or. Shade both conditions on one number line before you write the answer.
Where / symbols
- or
- A compound inequality joins two statements with or or and.
- union
- Or means union: a number is in the solution set if it works in at least one part.
- and
- A compound inequality joins two statements with or or and.
- intersection
- And means intersection: the number must work in both parts at once.
Overview
Union spreads out; intersection overlaps:
Or Means Union
Or is union: keep every number that satisfies at least one of the two parts. For x < 1 or x > 4 the solution set is two separate rays. There is no single interval that covers both. A number such as 2 sits in the gap and belongs to neither set, so it is out.
Examples x < 1 or x > 4 · x <= -2 or x >= 5 · x < 0 or x > 0 (all except 0)
And Means Intersection
And is intersection: keep only the overlap. For x >= 0 and x < 3 the solution is 0 <= x < 3. The chain -2 < x < 5 means x > -2 and x < 5, so it is AND. Both edges of that chain are open because both symbols are strict. If you wrote -2 <= x <= 5, both edges would be closed.
Examples -2 < x < 5 · 1 <= x <= 4 · x > -1 and x < 2
What it means
A compound inequality joins two statements with or or and.
Main ideas
1. OR MEANS UNION: Or is union: keep every number that satisfies at least one of the two parts. 2. AND MEANS INTERSECTION: And is intersection: keep only the overlap.
Why it matters
Read a chain from the middle: -2 < x < 5 says x is greater than -2 and less than 5. It does not mean -2 < 5, which is always true and useless. Graph the two walls, then shade only the segment between them for AND, or shade outward for or.
How to answer in an exam
Never replace a chain by or. -2 < x < 5 is intersection, not union. Or produces two regions; and produces the overlap. If the two and-conditions cannot overlap (x > 5 and x < 2), the solution set is empty.
Explanation
Read a chain from the middle: -2 < x < 5 says x is greater than -2 and less than 5. It does not mean -2 < 5, which is always true and useless. Graph the two walls, then shade only the segment between them for AND, or shade outward for or.
In simple words
Never replace a chain by or. -2 < x < 5 is intersection, not union. Or produces two regions; and produces the overlap. If the two and-conditions cannot overlap (x > 5 and x < 2), the solution set is empty.
Key points
- or
- union
- and
- intersection
- at least one
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 statements with or or and.
Key Terms
- or: A compound inequality joins two statements with or or and.
- union: Or means union: a number is in the solution set if it works in at least one part.
Main Pattern
- 1. OR MEANS UNION: Or is union: keep every number that satisfies at least one of the two parts.
- 2. AND MEANS INTERSECTION: And is intersection: keep only the overlap.
Worked Reminder
- x < 1 or x > 4
- x <= -2 or x >= 5
- x < 0 or x > 0 (all except 0)
Common Mistake
- Never replace a chain by or. -2 < x < 5 is intersection, not union. Or produces two regions; and produces the overlap. If the two and-conditions cannot overlap (x > 5 and x < 2), the solution set is empty.
Exam Tip
- State the rule for compound inequalities, show each algebraic step, and check the final answer against the question.
60-second revision
- A compound inequality joins two statements with or or and.
- Or is union: keep every number that satisfies at least one of the two parts.
- And is intersection: keep only the overlap.
- Read a chain from the middle: -2 < x < 5 says x is greater than -2 and less than 5.
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