Inequalities (Copy)

2.6 Inequalities – Cheat Sheet

1. Representing Inequalities on a Number Line

Example:
−3 ≤ x < 1 → Closed circle at −3, open circle at 1, shaded between them.

2. Solving and Representing Linear Inequalities in One Variable

Example 1:
3x < 2x + 4

• Subtract 2x: x < 4
• Number line: open circle at 4, shade left.

Example 2:
−3 ≤ 3x − 2 < 7

• Add 2: −1 ≤ 3x < 9
• Divide by 3: −1/3 ≤ x < 3
• Number line: closed circle at −1/3, open circle at 3.

3. Inequality Symbols Recap

4. Linear Inequalities in Two Variables (Graphical)

Conventions:

• Broken line: Strict inequalities (<, >)
• Solid line: Inclusive inequalities (≤, ≥)
• Shading: Represents unwanted region unless question asks for the wanted region to be shaded.

Example:
x < 1 → Broken vertical line at x = 1, shade left.
y ≥ 1 → Solid horizontal line at y = 1, shade above.

5. Listing Inequalities for a Given Region

1. Identify boundary lines (e.g., x = 2, y = −1, y = 3x + 4).
2. Check if lines are solid (≤, ≥) or broken (<, >).
3. Test a point in the shaded region to determine inequality sign.

Example:
Region inside rectangle bounded by:
x = 0, x = 2, y = 1, y = 3
Inequalities: 0 ≤ x ≤ 2, 1 ≤ y ≤ 3

6. Examples Table for Quick Reference