Inequalities and Number Line

Algebra-1

1. Fundamental Concepts

Definition: Inequalities are mathematical statements that use symbols such as $<$ , $>$ , $\leq$ , and $\geq$ to compare two expressions.
Number Line: A visual representation of numbers where each point on the line corresponds to a number.
Solution Set: The set of all values that satisfy an inequality.

2. Key Concepts

Basic Rule: $$x \cdot x = x^2$$

Degree Preservation: The highest degree in the result matches input

Application: Used to combine expressions in physics/engineering

3. Examples

Example 1 (Basic)

Problem: Graph the solution set for $$x > 3$$

Step-by-Step Solution:

Draw a number line and place an open circle at 3, indicating that 3 is not included in the solution set.
Shade the region to the right of 3, indicating all numbers greater than 3 are part of the solution set.

Validation: Check with a test point like x=4 → 4 > 3 ✓

Example 2 (Intermediate)

Problem: Solve and graph $$2x - 5 < 7$$

Step-by-Step Solution:

Add 5 to both sides: $$2x < 12$$
Divide by 2: $$x < 6$$

          Draw a number line and place an open circle at 6.          Shade the region to the left of 6.

Validation: Check with a test point like x=5 → 2(5) - 5 < 7; 10 - 5 < 7; 5 < 7 ✓

4. Problem-Solving Techniques

Visual Strategy: Use a number line to visualize the solution set.
Error-Proofing: Always check solutions with test points.
Concept Reinforcement: Practice solving inequalities with different operations.