Shading Inequalities- Graphing Techniques Explained
What Shading Inequalities Actually Means
When you graph an inequality, you're not just drawing a line. You're showing every possible point that satisfies the relationship. The shaded region is your solution set — every coordinate pair that makes the inequality true.
Most students get this wrong. They shade randomly or shade the wrong side entirely. Here's the actual method that works every time.
The Test Point Method — Your Only Reliable Tool
Forget "above the line" or "below the line." That rule fails constantly depending on your inequality sign and which variable is positive. Instead, use this:
- Graph the boundary line (equals sign)
- Pick any point not on that line
- Plug it into your inequality
- Shade the side where the test point makes the inequality true
That's it. The origin (0, 0) is usually your best choice — unless it's on your boundary line, in which case pick something else like (1, 0) or (0, 1).
Solid vs. Dashed Lines
This part trips people up:
- Solid line — the boundary is included in the solution (≤ or ≥)
- Dashed line — the boundary is NOT included (< or >)
A solid line says "points on this line count." A dashed line says "points on this line don't count." Don't mix these up or your graph will be technically wrong even if the shading looks right.
Linear Inequalities — Step by Step
Example: Graph y > 2x + 3
- Change to equals: y = 2x + 3
- Graph the line — dashed because it's >, not ≥
- Test point (0, 0): 0 > 2(0) + 3 → 0 > 3 → FALSE
- Since (0, 0) is false, shade the opposite side of the line
That's the whole process. The test point fails, so everything on that side is out. Shade the other side.
Common Mistake
Students often shade based on the slope direction — shading "above" because the line goes up. Wrong. The test point method works regardless of slope, y-intercept, or how tilted your line is. Test (0, 0) every time unless it's on the line.
System of Inequalities — Overlapping Regions
When you have multiple inequalities, you shade each one separately. Your solution is where all shaded regions overlap.
Example system:
- y ≥ x + 1
- y < -2x + 4
- Graph y ≥ x + 1 — solid line, test (0, 0): 0 ≥ 1 → FALSE, shade opposite
- Graph y < -2x + 4 — dashed line, test (0, 0): 0 < 4 → TRUE, shade that side
- Find where both shaded regions intersect
The final solution is the overlapping region. If there's no overlap, the system has no solution — that happens.
Special Cases That Break the Rules
Horizontal Boundaries
y > 3 means shade everything above the horizontal line y = 3. Test (0, 0): 0 > 3 is false, so shade above the line. This happens to match the "greater = above" rule, but that's coincidental, not a method.
Vertical Boundaries
x ≤ -2 means shade everything left of the vertical line x = -2. Test (0, 0): 0 ≤ -2 is false, so shade left. Again, this matches the intuitive direction, but it only works because of how the test point turned out.
When (0, 0) Is On the Line
If your boundary line passes through the origin, (0, 0) is useless as a test point. Pick something else. (1, 0), (0, 1), (1, 1) — anything not on the line works.
Quick Reference Table
| Inequality | Line Type | Test (0,0) Result | Shade Side |
|---|---|---|---|
| y > mx + b | Dashed | False | Opposite of (0,0) |
| y ≥ mx + b | Solid | False | Opposite of (0,0) |
| y < mx + b | Dashed | True | Same side as (0,0) |
| y ≤ mx + b | Solid | True | Same side as (0,0) |
How to Shade — Getting Started
Here's your workflow for any inequality:
- Rewrite with the equals sign to get your boundary equation
- Graph that line — solid or dashed based on your inequality sign
- Test the origin (0, 0) unless it's on the line
- Shade toward the test point if true, away if false
- Repeat for each inequality in a system, then find the overlap
That's the complete process. No guessing, no "which way does it go" confusion. Test point tells you everything.
Why Students Still Get This Wrong
The problem isn't the math. The problem is memorizing rules instead of understanding the method. "Greater than means shade above" works for simple cases and fails for everything else.
The test point method is universal. It works for horizontal lines, vertical lines, steep lines, shallow lines, positive slopes, negative slopes. Every single time.
Use it and stop guessing.