Solve One-Step Inequalities - Algebra-1

1. Fundamental Concepts

Definition: One-step inequalities are inequalities that can be solved in a single step by applying an arithmetic operation (addition, subtraction, multiplication, or division).
Properties of Inequality: The inequality sign changes direction when both sides are multiplied or divided by a negative number.

2. Key Concepts

Basic Rule: Isolating the Variable
Use the inverse operation to undo the constant or coefficient attached to the variable:
For , subtract from both sides: .
For , add to both sides: .
For , ( ), divide both sides by : .
For , ( ), divide both sides by : .

3. Examples

Example 1 (Basic)

Problem: Solve .

Step-by-Step Solution:

Add 7 to both sides to isolate : .

Validation: Substitute x = 12 → Original: 12 - 7 = 5 ≥ 4 ✓

Example 2 (Basic)

Problem: Solve .

Step-by-Step Solution:

Divide both sides by 5 (positive, so no symbol reversal): :

Validation: Substitute x = 5 → Original: 5(5) = 25 < 30 ✓

Example 3 (Intermediate)

Problem: Solve .

Step-by-Step Solution:

Multiply both sides by (negative, so reverse the symbol):

Validation: Substitute y = -33 → Original: 2 ✓

4. Problem-Solving Techniques

Identify the Inverse Operation: Match the operation in the inequality to its inverse (e.g., undo subtraction with addition).
Watch for Negative Multipliers/Divisors: Always reverse the inequality symbol when multiplying or dividing by a negative number!
Check Your Solution: Substitute a number from the solution set back into the original inequality to verify. (Example: For , test (true).