Variable on Both Sides - Inequalities - Algebra-1

1. Fundamental Concepts

Definition: Inequalities with variables on both sides are mathematical statements where the variable appears on both sides of the inequality symbol (>, <, ≥, ≤).
Objective: To solve such inequalities, isolate the variable on one side while maintaining the inequality's truth.
Properties: The properties of equality apply to inequalities, including addition, subtraction, multiplication, and division, but with special attention to the direction of the inequality when multiplying or dividing by a negative number.

Basic Rule: $${\text{If }} a > b \text{, then } a + c > b + c$$

Multiplication/Division by Negative: $${\text{If }} a > b \text{ and } c < 0, \text{ then } ac < bc$$

Solving Strategy: Move all terms involving the variable to one side and constants to the other.

Problem: Solve $$3x + 2 > 5x - 4$$

Validation: Substitute $x = 2$: Original: $3(2) + 2 > 5(2) - 4$ → $8 > 6$ ✓

Problem: Solve $$4y - 7 < 2y + 5$$

Validation: Substitute $y = 5$: Original: $4(5) - 7 < 2(5) + 5$ → $13 < 15$ ✓

Isolation Method: Always start by isolating the variable term on one side of the inequality.
Sign Change Awareness: Be cautious when multiplying or dividing by a negative number; reverse the inequality sign.
Verification Step: After solving, substitute a value from the solution set back into the original inequality to verify correctness.