📚 Understanding One-Step Inequalities
One-step inequalities are inequalities that require only one operation to isolate the variable. This operation can be addition, subtraction, multiplication, or division.
- ➕ Addition: $x - 3 > 5$ requires adding 3 to both sides.
- ➖ Subtraction: $x + 2 < 7$ requires subtracting 2 from both sides.
- ✖️ Multiplication: $\frac{x}{4} \geq 2$ requires multiplying both sides by 4.
- ➗ Division: $4x \leq 16$ requires dividing both sides by 4.
🧮 Understanding Multi-Step Inequalities
Multi-step inequalities, as the name suggests, require two or more operations to isolate the variable. These often involve a combination of distribution, combining like terms, and then performing addition/subtraction and multiplication/division.
- ➕➖ Combining Like Terms: $2x + 3x - 5 > 10$ requires combining $2x$ and $3x$ first.
- ↔️ Distribution: $3(x + 2) < 15$ requires distributing the 3 across $(x + 2)$.
- 🔢 Multiple Operations: $2x + 5 \leq 9$ requires subtracting 5 and then dividing by 2.
📊 Comparison Table: One-Step vs. Multi-Step Inequalities
| Feature |
One-Step Inequalities |
Multi-Step Inequalities |
| Number of Operations |
One |
Two or more |
| Complexity |
Simpler |
More complex |
| Steps Involved |
Isolate the variable in one step |
Involves distribution, combining like terms, and/or multiple operations |
| Example |
$x + 4 > 7$ |
$2(x - 1) < 8$ |
🔑 Key Takeaways
- 💡 One-step inequalities are solved with a single operation, making them quicker to solve.
- 🧠 Multi-step inequalities require multiple steps, including distribution or combining like terms, before isolating the variable.
- ✍️ Understanding the order of operations is crucial for solving multi-step inequalities correctly. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).