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 $$x + a \lt b$$ , subtract $$a$$ from both sides: $$x \lt b - a$$ .
For $$x - a \gt b$$ , add $$a$$ to both sides: $$x \gt b + a$$ .
For $$ax \geq b$$ , ( $$a \gt 0$$ ), divide both sides by $$a$$ : $$x \geq \frac{b}{a}$$ .
For $$ax \leq b$$ , ( $$a \lt 0$$ ), divide both sides by $$a$$ : $$x \geq \frac{b}{a}$$ .

3. Examples

Example 1 (Basic)

Problem: Solve $$x-7\geq4$$ .

Step-by-Step Solution:

Add 7 to both sides to isolate $$x$$ : $$x-7+7 \geq 4+7 \Rightarrow x\geq11$$ .

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

Example 2 (Basic)

Problem: Solve $$5x\lt30$$ .

Step-by-Step Solution:

Divide both sides by 5 (positive, so no symbol reversal): $$\frac{5x}{5}\lt\frac{30}{5}\Rightarrow x \lt 6$$ :

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

Example 3 (Intermediate)

Problem: Solve $$\frac{y}{-3}\leq12$$ .

Step-by-Step Solution:

Multiply both sides by $$-3$$ (negative, so reverse the symbol): $$\frac{y}{-3}\times(-3)\geq12\times(-3) \Rightarrow y\geq-36$$

Validation: Substitute y = -33 → Original: 2 $$\frac{-33}{-3}=11\leq12$$ ✓

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 $$5x \lt 30 \Rightarrow x \lt 6$$ , test $$x=5: 5(5)=25\lt30$$ (true).