Solve Compound Inequality - Algebra-1

1. Fundamental Concepts

Definition: Compound inequalities are statements that combine two or more inequalities using the words "and" or "or".
Graphical Representation: Solutions to compound inequalities can be represented on a number line.
Types: There are two types of compound inequalities: conjunctions (using "and") and disjunctions (using "or").

2. Key Concepts

Basic Rule: $$(x \geq a) \text{ and } (x \leq b) \implies a \leq x \leq b$$

Degree Preservation: The solution set for a compound inequality is the intersection or union of the individual solutions, depending on whether it's an "and" or "or" statement.

Application: Used in real-world scenarios such as determining ranges of acceptable values in engineering or economics.

3. Examples

Example 1 (Basic)

Problem: Solve the compound inequality $$(x + 3) > 5 \text{ and } (x - 2) < 4$$

Step-by-Step Solution:

Solve each inequality separately:
- $x + 3 > 5 \implies x > 2$
- $x - 2 < 4 \implies x < 6$
Combine the solutions since it's an "and" statement: $2 < x < 6$

Validation: Substitute $x = 4$ → Original: $(4 + 3) > 5$ and $(4 - 2) < 4$; Simplified: $7 > 5$ and $2 < 4$ ✓

Example 2 (Intermediate)

Problem: Solve the compound inequality $$(2x - 1) \geq 3 \text{ or } (x + 5) \leq 8$$

Step-by-Step Solution:

Solve each inequality separately:
- $2x - 1 \geq 3 \implies 2x \geq 4 \implies x \geq 2$
- $x + 5 \leq 8 \implies x \leq 3$
Combine the solutions since it's an "or" statement: $x \geq 2 \text{ or } x \leq 3$

Validation: Substitute $x = 2$ → Original: $(2(2) - 1) \geq 3$ or $(2 + 5) \leq 8$; Simplified: $3 \geq 3$ or $7 \leq 8$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use a number line to represent the solution sets visually.
Error-Proofing: Always check the validity of the solution by substituting values back into the original inequalities.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding of both "and" and "or" compound inequalities.