Inequalities
Shade above or below? Dashed or solid line? Learn the visual language of SAT Inequalities.
Foundation: Reading the Symbols
Inequalities are just equations with a "direction".
Strict
"Less than" / "Greater than"
Inclusive
"...or equal to"
The Shading Cheat Sheet
Assuming \(y\) is on the left side (e.g., \(y > ...\)):
> or ≥
SHADE ABOVE
Think: "Greater" = "Higher"
< or ≤
SHADE BELOW
Think: "Less" = "Lower"
2D Linear Inequalities
On the Cartesian plane, an inequality represents an entire region of solutions.
Example: \(y > 3x + 2\)
- 📈 Line: \(y = 3x + 2\) (Dashed)
- 🎨 Shading: Above the line
- 💡 Logic: "Greater than" means we want the region with higher y-values.
Example: \(y \le -x + 4\)
- 📉 Line: \(y = -x + 4\) (Solid)
- 🎨 Shading: Below the line
- 💡 Logic: "Less than or equal to" includes the line and everything under it.
| Symbol | Line Type | Meaning |
|---|---|---|
| > or < | Dashed | Points ON the line are NOT solutions. |
| ≥ or ≤ | Solid | Points ON the line ARE solutions. |
"Which point is a solution?" If the point is ON a dashed line, the answer is NO. If it's ON a solid line, the answer is YES.
The Golden Rule
When to FLIP the sign. If you divide by a negative, the world turns upside down.
See Rules →Desmos Shading
Don't guess. Type it in. Desmos handles dashed lines and shading automatically.
View Desmos Hack →The "NOT" Trap
"Which point is NOT a solution?" Why boundary lines matter.
Spot the Trap →