Inequalities Introduction - Algebra-1

1. Fundamental Concepts

Definition: Inequalities are mathematical statements that compare two expressions using symbols such as $$\lt$$ , $$\gt$$ , $$\leq$$ , and $$\geq$$ .
Symbols:
- $$\lt$$ means "less than"
- $$\gt$$ means "greater than"
- $$\leq$$ means "less than or equal to"
- $$\geq$$ means "greater than or equal to"
Basic Properties: Inequalities can be manipulated by adding, subtracting, multiplying, or dividing both sides by the same number, with special rules for multiplication and division by negative numbers.

Basic Rule: If $$a \textless b$$ , then $$a + c \textless b + c$$

Multiplication by Negative Numbers: If $$a \textless b$$ and $$c \textless 0$$ , then $$ac \textgreater bc$$

Solving Inequalities: Isolate the variable on one side of the inequality

Problem: Solve the inequality $$2x - 3 \textless 5$$

Validation: Substitute $$x = 3$$ → Original: $$2(3) - 3 \textless 5$$ ; Simplified: $$6 - 3 \textless 5$$ ✓

Problem: Solve the inequality $$-4x + 7 \geq 15$$

Validation: Substitute $$x = -3$$ → Original: $$-4(-3) + 7 \geq 15$$ ; Simplified: $$12 + 7 \geq 15$$ ✓

Isolation Strategy: Always isolate the variable on one side of the inequality.
Sign Reversal Awareness: Be cautious when multiplying or dividing by a negative number; remember to reverse the inequality sign.
Graphical Representation: Use number lines to visualize solutions and understand the range of values that satisfy the inequality.