MathematicsChapter 4: Linear and Quadratic Inequalities
Properties of Inequalities
Trichotomy: for reals a and b, exactly one of a<b, a=b, a>b holds.
Learning objectives
- 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.
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 positive keeps it. Multiplying (or dividing) by a negative reverses it. That reverse is the chapter's most examined move.
Where / symbols
- 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.
Overview
Keep or flip:
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.
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.
Examples -3x>=39 -> x<=-13 · Times -1 flips · Do not flip for +c
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.
Worked example
- solve 17 - 3x >= 56.
- 17 - 3x >= 56
- -3x >= 56 - 17
- -3x >= 39
- divide by -3 and reverse the inequality
- x <= -13
In simple words
The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip there.
Key points
- add: same direction
- times negative: reverse
- direction stays
- flip when negative
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Properties of Inequalities
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- Worked: solve 17 - 3x >= 56.
- 17 - 3x >= 56
- -3x >= 56 - 17
- -3x >= 39
- divide by -3 and reverse the inequality
- x <= -13
Common Misconception
- The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip there.
FBISE Exam Check
- The flip happens only for multiplication or division by a negative. Subtracting a negative is just adding; do not flip there.
60-second revision
- 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
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