Properties of Inequalities

Properties of Inequalities

What is it?

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.

In simple words

• 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.

Important 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.

positive: Multiply or divide by a positive number and the direction keeps.

keeps: Multiply or divide by a positive number and the direction keeps.

What you should learn

• Explain properties of inequalities in clear mathematical language
• Apply the ideas in additive and positive
• Solve problems involving negative reverses

Examples

• x-4 < 2 becomes x < 6
• 3x < 12 becomes x < 4
• x/2 >= 5 becomes x >= 10

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.

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.

Full school-level explanation

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.

Direction is the whole story:

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.

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.

Remember

• additive property
• keeps direction
• positive number

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.

Remember

• negative
• REVERSES
• x < -3

Examples

• -2x > 6 becomes x < -3
• -x <= 4 becomes x >= -4
• 5-x > 1 becomes x < 4

Remember

• Equivalent inequalities
• additive property
• positive
• keeps
• negative
• REVERSES
• hidden negative
• same solution set

How the ideas connect

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.

Deeper understanding

Go beyond memorising Properties of Inequalities. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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.

Common mistakes

Exam focus

Core Idea

Key Terms

Main Pattern

Worked Reminder

Exam focus

Quick revision

• Equivalent inequalities
• additive property
• positive
• keeps
• 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

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.

For long answers on Properties of Inequalities: define clearly → explain the main parts → give one example → state one exam caution.