Flip the Inequality Sign

Algebra-1

1. Fundamental Concepts

Definition: Inequalities are mathematical statements that use symbols such as <, >, ≤, and ≥ to compare two expressions.
Inequality Symbols:
- < means "less than"
- > means "greater than"
- ≤ means "less than or equal to"
- ≥ means "greater than or equal to"
Flipping the Inequality Sign: The inequality sign must be flipped when multiplying or dividing both sides of an inequality by a negative number.

2. Key Concepts

Basic Rule: $-a \cdot b \lt -a \cdot c$ implies $b \gt c$ if $a \gt 0$

Degree Preservation: The direction of the inequality changes when multiplied or divided by a negative number

Application: Used in solving real-world problems involving constraints and comparisons

3. Examples

Example 1 (Basic)

Problem: Solve the inequality $-2x + 4 \gt 6$

Step-by-Step Solution:

Subtract 4 from both sides: $-2x \gt 2$
Divide both sides by -2 and flip the inequality sign: $x \lt -1$

Validation: Substitute x = -2 → Original: -2(-2) + 4 > 6; Simplified: 4 + 4 > 6 ✓

Example 2 (Intermediate)

Problem: Solve the inequality $5 - 3y \leq 14$

Step-by-Step Solution:

Subtract 5 from both sides: $-3y \leq 9$
Divide both sides by -3 and flip the inequality sign: $y \geq -3$

Validation: Substitute y = -3 → Original: 5 - 3(-3) ≤ 14; Simplified: 5 + 9 ≤ 14 ✓

4. Problem-Solving Techniques

Visual Strategy: Use a number line to visualize the solution set of inequalities.
Error-Proofing: Always check the direction of the inequality sign after performing operations with negative numbers.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding of when to flip the inequality sign.