Mastering Linear Inequalities: The Ultimate Algebra 1 Guide
⚠️ CRITICAL WARNING: The "Sign-Flip" & "Distribution" Traps
Trap 1: 90% of students fail to distribute the number outside the parentheses to every term inside. Don't leave the second term behind!
Trap 2: If you multiply or divide by a negative number, you MUST flip the inequality sign. Forgetting this turns a correct answer into a total disaster. Read every step below.
1. What is an Inequality? (Understanding the "Range")
Unlike an equation (e.g., x = 5), where the answer is a single point on a map, an Inequality represents a Range of Solutions. It’s like saying, "You need at least $20 to buy this game." $20 is the start, but $21, $30, or $100 all work. That "infinite range" is what we represent on a number line.
2. Advanced Example: The Double-Negative Challenge
Let's solve: -3(x - 4) ≥ 15
Step 1: Distribution (Watch the signs!)
-3 * x = -3x
-3 * -4 = +12 (Crucial: Negative times negative is positive!)
Equation: -3x + 12 ≥ 15
Step 2: Isolation
Subtract 12 from both sides: -3x ≥ 3
Step 3: The Flip
Divide by -3. Since we divide by a negative, we flip the sign from (≥) to (≤):
x ≤ -1
3. Comparison Table: Equation vs. Inequality
| Feature | Equation (=) | Inequality (<, >) |
|---|---|---|
| Solution Type | Specific point | Infinite range |
4. Solution to the Challenge
Challenge: -3(x - 2) ≥ 12
Solution Breakdown:
- Distribute: -3x + 6 ≥ 12
- Subtract 6: -3x ≥ 6
- Divide by -3 (Flip the sign!): x ≤ -2
The solution is any number less than or equal to -2. On a number line, you draw a closed circle at -2 and shade to the left.
5. Frequently Asked Questions (FAQ)
Q: When do I use an open circle vs. a closed circle?
A: Use an open circle for < or > (the number itself is not included). Use a closed circle for ≤ or ≥ (the number is included).
