MathematicsChapter 5: Linear Equations and Inequalities
Properties of Inequalities
Equivalent inequalities come from legal moves, just as equivalent equations do, with one extra rule.
Learning objectives
- Explain properties of inequalities in clear mathematical language
- Apply the ideas in additive and positive
- Solve problems involving negative reverses
Definition
Equivalent inequalities come from legal moves, just as equivalent equations do, with one extra rule. The additive property says you may add or subtract the same number on both sides and the inequality direction stays the same. Multiply or divide by a positive number and the direction keeps. Multiply or divide by a negative number and the direction REVERSES. Forgetting the reverse is the most common Class 9 trap.
Where / symbols
- Equivalent inequalities
- Equivalent inequalities come from legal moves, just as equivalent equations do, with one extra rule.
- additive property
- The additive property says you may add or subtract the same number on both sides and the inequality direction stays the same.
- positive
- Multiply or divide by a positive number and the direction keeps.
- keeps
- Multiply or divide by a positive number and the direction keeps.
Overview
Direction is the whole story:
Additive and Positive
The additive property keeps direction: from x-4 < 2 add 4 to get x < 6. Multiply or divide by a positive number and the symbol is unchanged: 3x < 12 divide by 3 gives x < 4. Adding a negative number is still addition, so the direction still keeps. Only a negative multiplier or divisor flips the symbol.
Examples x-4 < 2 becomes x < 6 · 3x < 12 becomes x < 4 · x/2 >= 5 becomes x >= 10
Negative Reverses
When you multiply or divide by a negative, the inequality REVERSES. From -2x > 6 divide by -2 and switch to x < -3. Check: x=-4 gives -2(-4)=8 > 6, true; x=-2 gives -2(-2)=4 which is not greater than 6. The same reverse happens for <, <=, and >=. Recite it: negative REVERSES.
Examples -2x > 6 becomes x < -3 · -x <= 4 becomes x >= -4 · 5-x > 1 becomes x < 4
What it means
Equivalent inequalities come from legal moves, just as equivalent equations do, with one extra rule.
Main ideas
1. ADDITIVE AND POSITIVE: The additive property keeps direction: from x-4 < 2 add 4 to get x < 6. 2. NEGATIVE REVERSES: When you multiply or divide by a negative, the inequality REVERSES.
Why it matters
A hidden negative appears when you swap sides of an inequality, or when you multiply by -1 to make the x term positive. In 5-x > 1, subtracting 5 then multiplying by -1 both need care; many students move x across and forget the reverse. Equivalent inequalities must have the same solution set after every step.
How to answer in an exam
Equations never reverse when you multiply by a negative. Inequalities do. If a is negative in ax+b < 0, dividing by a REVERSES the symbol. Write the new symbol on the same line as the division so you cannot skip it.
Explanation
A hidden negative appears when you swap sides of an inequality, or when you multiply by -1 to make the x term positive. In 5-x > 1, subtracting 5 then multiplying by -1 both need care; many students move x across and forget the reverse. Equivalent inequalities must have the same solution set after every step.
In simple words
Equations never reverse when you multiply by a negative. Inequalities do. If a is negative in ax+b < 0, dividing by a REVERSES the symbol. Write the new symbol on the same line as the division so you cannot skip it.
Key points
- Equivalent inequalities
- additive property
- positive
- keeps
- keeps direction
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
- Equivalent inequalities come from legal moves, just as equivalent equations do, with one extra rule.
Key Terms
- Equivalent inequalities: Equivalent inequalities come from legal moves, just as equivalent equations do, with one extra rule.
- additive property: The additive property says you may add or subtract the same number on both sides and the inequality direction stays the same.
Main Pattern
- 1. ADDITIVE AND POSITIVE: The additive property keeps direction: from x-4 < 2 add 4 to get x < 6.
- 2. NEGATIVE REVERSES: When you multiply or divide by a negative, the inequality REVERSES.
Worked Reminder
- x-4 < 2 becomes x < 6
- 3x < 12 becomes x < 4
- x/2 >= 5 becomes x >= 10
Common Mistake
- Equations never reverse when you multiply by a negative. Inequalities do. If a is negative in ax+b < 0, dividing by a REVERSES the symbol. Write the new symbol on the same line as the division so you cannot skip it.
Exam Tip
- State the rule for properties of inequalities, show each algebraic step, and check the final answer against the question.
60-second revision
- Equivalent inequalities come from legal moves, just as equivalent equations do, with one extra rule.
- The additive property keeps direction: from x-4 < 2 add 4 to get x < 6.
- When you multiply or divide by a negative, the inequality REVERSES.
- A hidden negative appears when you swap sides of an inequality, or when you multiply by -1 to make the x term positive.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.