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Induction On Inequalities
Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
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Induction On Inequalities
What is it?
Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
In simple words
- Assume P(k), multiply both sides by a positive number, or add a positive term.
- Multiply by (1+x) if x>-1
- Show extra >= 0
- State the rule, show the working, and check the answer.
Important terms
INDUCTION ON INEQUALITIES: Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
INEQUALITIES: Assume P(k), multiply both sides by a positive number, or add a positive term.
What you should learn
- State the rule or meaning of induction on inequalities in easy words.
- Apply induction on inequalities in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Examples
- Multiply by (1+x) if x>-1
- Show extra >= 0
- End at P(k+1)
MAIN IDEA
Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
EXAM USE
Write the rule first: Assume P(k), multiply both sides by a positive number, or add a positive term. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Recommended for Class 11
Full school-level explanation
Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
Understand the idea, then apply it:
WHAT — INDUCTION ON INEQUALITIES
Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
WHY — Why it matters
This rule is a separate learning unit inside induction on divisibility and inequalities. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Assume P(k), multiply both sides by a positive number, or add a positive term. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
MAIN IDEA
Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
Remember
- keep the inequality direction
Examples
- Multiply by (1+x) if x>-1
- Show extra >= 0
- End at P(k+1)
EXAM USE
Write the rule first: Assume P(k), multiply both sides by a positive number, or add a positive term. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Remember
- focused answer
- accurate support
Examples
- State the central idea
- Use one accurate detail
- Check the final response
How to answer a induction on inequalities question
How it works
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
Remember
- keep the inequality direction
- write the k+1 expression in terms of the k piece
- leftover must be divisible / positive
- split into IH piece + 3
- keep the inequality direction
- focused answer
- accurate support
- keep the inequality direction
Deeper understanding
Go beyond memorising Induction On Inequalities. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
What to Know
Example or Evidence
How to Answer
Exam focus
Quick revision
- keep the inequality direction
- write the k+1 expression in terms of the k piece
- leftover must be divisible / positive
- split into IH piece + 3
- INDUCTION ON INEQUALITIES: Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction o
- INEQUALITIES: Assume P(k), multiply both sides by a positive number, or add a positive term.
Write 4k+1-1 as 4(4k-1)+3 so both pieces are visibly multiples of 3.
For long answers on Induction On Inequalities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- keep the inequality direction
- write the k+1 expression in terms of the k piece
- leftover must be divisible / positive
- split into IH piece + 3
Important formulas
- keep the inequality direction
- write the k+1 expression in terms of the k piece
- leftover must be divisible / positive
Common exam traps
- Write 4k+1-1 as 4(4k-1)+3 so both pieces are visibly multiples of 3.
- Do not choose a formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write 4k+1-1 as 4(4k-1)+3 so both pieces are visibly multiples of 3.
- Mistake
Do not choose a formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes, restrictions, or skipped algebra steps.
- Mistake
Write 4k+1-1 as 4(4k-1)+3 so both pieces are visibly multiples of 3.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
keep the inequality direction
write the k+1 expression in terms of the k piece
leftover must be divisible / positive
split into IH piece + 3
Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
- Definition
- Assume P(k), multiply both sides by a positive number, or add a positive term. Track the direction of >= . Base often n=1 or n=2 as stated.
- Example
- Multiply by (1+x) if x>-1
- Critical warning
- Write 4k+1-1 as 4(4k-1)+3 so both pieces are visibly multiples of 3.
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Exam Practice
Define
Q1. Define Induction On Inequalities.
Explain
Q2. Explain: MAIN IDEA.
Example
Q3. Give one clear example of Induction On Inequalities and say why it fits.
Mistake
Q4. State one common mistake students make about Induction On Inequalities.
Exam note
Q5. Write a short exam note on Induction On Inequalities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.