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Properties of Inequalities
Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the inequality direction. Multiplying (or dividing) by a positive keeps it. Multiplying (or dividing) by a negative reverses it. That reverse is the chapter's most examined move.
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Properties of Inequalities
What is it?
Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the inequality direction. Multiplying (or dividing) by a positive keeps it. Multiplying (or dividing) by a negative reverses it. That reverse is the chapter's most examined move.
In simple words
- Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds.
- ADDITION AND POSITIVE SCALE: If a>b then a+c>b+c.
- NEGATIVE SCALE: If a<b and c<0 then ac>bc.
- Worked-method reminder: x <= -13
Important terms
PROPERTIES OF INEQUALITIES: Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the inequality direction. Multiplying (or dividing) by a positive keeps it. Multiplying (or dividing) by a negative reverses it. That reverse is the chapter's most examined move.
ADDITION AND POSITIVE SCALE: If a>b then a+c>b+c. If a<b and c>0 then ac<bc. These feel like equations.
NEGATIVE SCALE: If a<b and c<0 then ac>bc. Dividing by -3 flips. -3x >= 39 becomes x <= -13. Forget the flip and the answer is the other ray.
What you should learn
- Explain properties of inequalities using the correct definitions, conditions, and notation.
- Apply add: same direction, times negative: reverse correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x+3<7 -> x<4
- 2x<10 -> x<5
- Same as 'do it to both sides'
ADDITION AND POSITIVE SCALE
If a>b then a+c>b+c. If a<b and c>0 then ac<bc. These feel like equations.
NEGATIVE SCALE
If a<b and c<0 then ac>bc. Dividing by -3 flips. -3x >= 39 becomes x <= -13. Forget the flip and the answer is the other ray.
Recommended for Class 10
Full school-level explanation
Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the inequality direction. Multiplying (or dividing) by a positive keeps it. Multiplying (or dividing) by a negative reverses it. That reverse is the chapter's most examined move.
Keep or flip:
WHAT — PROPERTIES OF INEQUALITIES
Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the inequality direction. Multiplying (or dividing) by a positive keeps it. Multiplying (or dividing) by a negative reverses it. That reverse is the chapter's most examined move. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Properties Of 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 ADDITION AND POSITIVE SCALE, then NEGATIVE SCALE. Worked: solve 17 - 3x >= 56. 17 - 3x >= 56 -3x >= 56 - 17 -3x >= 39 divide by -3 and reverse the inequality x <= -13 Finish by checking signs, restrictions, units, and whether the answer matches the question.
ADDITION AND POSITIVE SCALE
If a>b then a+c>b+c. If a<b and c>0 then ac<bc. These feel like equations.
Remember
- direction stays
Examples
- x+3<7 -> x<4
- 2x<10 -> x<5
- Same as 'do it to both sides'
NEGATIVE SCALE
If a<b and c<0 then ac>bc. Dividing by -3 flips. -3x >= 39 becomes x <= -13. Forget the flip and the answer is the other ray.
Remember
- flip when negative
Examples
- -3x>=39 -> x<=-13
- Times -1 flips
- Do not flip for +c
Remember
- add: same direction
- times negative: reverse
- x <= -13
- only times/divide negative
- direction stays
- flip when negative
- add: same direction
- times negative: reverse
How the ideas connect
Worked: solve 17 - 3x >= 56. 17 - 3x >= 56 -3x >= 56 - 17 -3x >= 39 divide by -3 and reverse the inequality x <= -13
Deeper understanding
Go beyond memorising Properties of Inequalities. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve 17 - 3x >= 56. 17 - 3x >= 56 -3x >= 56 - 17 -3x >= 39 divide by -3 and reverse the inequality x <= -13
Common mistakes
Exam focus
Core Idea
ADDITION AND POSITIVE SCALE
NEGATIVE SCALE
Worked Method
Exam focus
Quick revision
- add: same direction
- times negative: reverse
- x <= -13
- only times/divide negative
- PROPERTIES OF INEQUALITIES: Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the
- ADDITION AND POSITIVE SCALE: If a>b then a+c>b+c. If a<b and c>0 then ac<bc. These feel like equations.
The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip there.
For long answers on Properties of Inequalities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- add: same direction
- times negative: reverse
- x <= -13
- only times/divide negative
Important formulas
- add: same direction
- times negative: reverse
- x <= -13
Common exam traps
- The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip there.
- Do not flip for +c
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip there.
- Mistake
Do not flip for +c
- Mistake
Do not treat ADDITION AND POSITIVE SCALE and NEGATIVE SCALE as the same step; each has a different purpose.
- Mistake
The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip there.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
add: same direction
times negative: reverse
x <= -13
only times/divide negative
Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the inequality direction. Multiplying (or dividing) by a positiv
x <= -13
- Definition
- Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds. Adding the same number keeps the inequality direction. Multiplying (or dividing) by a positiv
- Formula
- x <= -13
- Example
- x+3<7 -> x<4
- Critical warning
- The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip t
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Exam Practice
Define
Q1. Define Properties of Inequalities.
Explain
Q2. Explain: ADDITION AND POSITIVE SCALE.
Example
Q3. Give one clear example of Properties of Inequalities and say why it fits.
Mistake
Q4. State one common mistake students make about Properties of Inequalities.
Exam note
Q5. Write a short exam note on Properties of Inequalities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.