Which Side to Shade- Inequality Graphing Made Easy
Which Side to Shade: The Inequality Graphing Problem, Solved
Graphing inequalities trips up more students than almost any other algebra skill. You nail the line, then you're stuck staring at two regions, wondering which one gets the shading. It's a dumb way to lose points on a test.
Here's the fix.
The Core Concept in 30 Seconds
A linear inequality like y > 2x + 1 divides the coordinate plane into two regions. One region contains all the points that make the inequality true. The other contains points that make it false.
Your job: figure out which region is which, then shade the correct one.
The boundary line itself is either solid or dashed. A solid line means the line is included in the solution (≤ or ≥). A dashed line means it's not included (< or >).
The Test Point Method: Always Works, Never Fails
There's exactly one reliable method that works for every inequality, no matter how complicated. It's called the test point method, and it's what you should use every single time.
How It Works
Pick any point that's not on the line. The origin (0, 0) is usually the easiest choice—unless the line happens to pass through it.
Plug those coordinates into the inequality. Check if the inequality holds true.
If it's true, shade the region containing that point. If it's false, shade the opposite region.
Example in Action
Let's shade y ≤ -x + 3.
First, draw the line y = -x + 3 as solid (because we have ≤).
Pick (0, 0) as our test point. Plug it in: 0 ≤ -0 + 3 → 0 ≤ 3. That's true.
So shade the region containing (0, 0)—the region below and to the right of the line.
When NOT to Use (0, 0)
If your line passes through the origin, (0, 0) is on the boundary. You can't use it. Pick something else like (1, 0) or (0, 1).
Same deal if the line is vertical (like x = 2) or horizontal (like y = -1). Just grab any point on the opposite side of the line from the origin.
Quick Visual Rules (Use as a Backup)
Once you've done enough of these, you'll notice patterns. Here are shortcuts—but only use them after you understand the test point method.
- y > mx + b → shade above the line
- y < mx + b → shade below the line
- y = mx + b is always the boundary
Watch out for inequalities solved for x instead of y, like x + 2y > 6. Rearrange it to isolate y first, then apply the rules above.
Line Type Reference
Your inequality symbol tells you whether the line is solid or dashed. Here's the breakdown:
| Symbol | Line Type | Included in Solution? |
|---|---|---|
| < and > | Dashed | No |
| ≤ and ≥ | Solid | Yes |
Step-by-Step: Graphing an Inequality from Scratch
Let's do a complete example: 2x - 3y > 12
Step 1: Rewrite as an equation
Replace > with = to get your boundary line: 2x - 3y = 12
Step 2: Find two points on the line
Set x = 0 → -3y = 12 → y = -4. So (0, -4) is on the line.
Set y = 0 → 2x = 12 → x = 6. So (6, 0) is on the line.
Plot both points and draw a dashed line through them (because we have >, not ≥).
Step 3: Pick a test point
(0, 0) is not on the line. Plug it in: 2(0) - 3(0) > 12 → 0 > 12. That's false.
Step 4: Shade accordingly
Since (0, 0) makes the inequality false, shade the region opposite to where (0, 0) sits—which means shading to the left and above the line.
Common Mistakes That Cost Points
- Shading the wrong side because they forgot to test the point. Just test it. Every time.
- Solid line when it should be dashed. Remember: < and > get dashed lines. ≤ and ≥ get solid lines.
- Forgetting to rearrange when y isn't isolated. You can't apply the "above/below" shortcuts until you solve for y first.
System of Inequalities
If you're graphing a system with two inequalities, you shade the region that satisfies both. The overlapping region is your solution. That's it.
Sometimes one inequality shades up, the other shades down, and you're left with a thin strip between two lines. That's correct. Don't second-guess it.
The Bottom Line
Stop guessing. Test a point. Plug (0, 0) into the inequality—if it's true, shade that side. If it's false, shade the other side. Solid or dashed line? Check the symbol. That's the entire process.
No memorization tricks. No "greater than means up" nonsense that falls apart the second the equation has a negative slope. Just test, shade, done.