Graphing Inequalities- Which Half Plane to Shade

Graphing Inequalities: Which Half Plane to Shade

Most students freeze up when they see a linear inequality and have to graph it. They get the line right, but then they're stuck staring at the coordinate plane, unsure which side to shade.

Here's the thing: there's a simple method that works every time. No guessing. No "I think it's this side." Once you learn it, you'll never hesitate again.

What You're Actually Looking At

When you graph a linear inequality like y > 2x + 1, you're not just drawing a line. You're dividing the entire coordinate plane into two regions. One region contains all the points that make the inequality true. The other contains points that make it false.

The line itself is the boundary. The shading shows which region satisfies your inequality.

The Test Point Method (Use This, Always)

Forget trying to memorize rules about "above" or "below" the line. Use the test point method. It works for every inequality, no exceptions.

Step 1: Graph the boundary line

• Convert the inequality to an equation: replace the inequality sign with an equals sign
• y > 2x + 1 becomes y = 2x + 1
• Use a dashed line if the inequality is strict (> or <)
• Use a solid line if the inequality includes equality (≥ or ≤)

Step 2: Pick a test point

Pick any point not on the line. The origin (0, 0) is usually the easiest, unless it happens to fall on your boundary line.

Step 3: Plug it in

Substitute the coordinates into your original inequality. See if the statement is true or false.

Step 4: Shade accordingly

• If the test point makes the inequality true, shade the region containing that point
• If the test point makes the inequality false, shade the opposite region

Why the Test Point Works

Think about it. The inequality y > 2x + 1 asks: "Which y-values are greater than 2x + 1?" Pick any x-value, like x = 0. Then 2(0) + 1 = 1. Points with y greater than 1 are above the line. Points with y less than 1 are below the line.

The test point just tells you which of those two regions satisfies your specific inequality.

Dashed vs Solid Lines

This trips people up constantly.

• Strict inequalities (y > or y <): The boundary line itself is not part of the solution. Use a dashed line.
• Inclusive inequalities (y ≥ or y ≤): The boundary line is part of the solution. Use a solid line.

Example: y ≥ x + 2 gets a solid line because points on the line work. y > x + 2 gets a dashed line because points on the line don't work.

Common Mistakes That Sabotage Students

Shading the wrong side

This happens when students try to guess based on the direction of the inequality symbol. They see ">" and assume "shade above." This works sometimes, but not when the line has negative slope or when x is isolated. The test point method removes all ambiguity.

Using a solid line for strict inequalities

Your graph will be technically incorrect. The boundary line should not be included in the solution set for > or <. A dashed line signals that.

Forgetting to convert the inequality sign

When graphing the boundary, you must replace the inequality with an equals sign. y > 3x - 2 becomes y = 3x - 2 for the line. Don't try to graph y > as if it's a function.

Picking a point on the line as your test point

That defeats the purpose. The line is the boundary between regions. A point on the line doesn't tell you anything about either side. Choose something clearly in one region or the other.

Examples

Example 1: y ≤ -x + 3

1. Boundary line: y = -x + 3
2. Line type: solid (≤ includes equality)
3. Test point: (0, 0)
4. Plug in: 0 ≤ -0 + 3 → 0 ≤ 3 → TRUE
5. Shade the region containing (0, 0)

The origin is below and to the left of the line. Shade that half-plane.

Example 2: y > ½x - 2

1. Boundary line: y = ½x - 2
2. Line type: dashed (> is strict)
3. Test point: (0, 0)
4. Plug in: 0 > ½(0) - 2 → 0 > -2 → TRUE
5. Shade the region containing (0, 0)

Since the inequality is strict, the boundary line stays dashed. Shade above the line this time.

Example 3: 2x + 3y ≤ 6

1. Rewrite in y = form: 3y ≤ -2x + 6 → y ≤ -⅔x + 2
2. Boundary line: y = -⅔x + 2
3. Line type: solid (≤)
4. Test point: (0, 0)
5. Plug in: 0 ≤ -⅔(0) + 2 → 0 ≤ 2 → TRUE
6. Shade the region containing (0, 0)

When x or y isn't isolated, solve for y first. Then apply the same test point method.

When the Origin Doesn't Work

Sometimes your boundary line passes through (0, 0). In that case, you can't use the origin as your test point. Pick any other point not on the line.

Example: y > 2x. The boundary line goes through (0, 0). Test point: (1, 0)

• Plug in: 0 > 2(1) → 0 > 2 → FALSE
• Shade the opposite side of the line

Quick Reference Table

Getting Started: Your Action Steps

1. Identify the boundary equation — replace the inequality sign with equals
2. Determine line type — dashed for strict, solid for inclusive
3. Graph the line using slope-intercept or any method you prefer
4. Pick (0, 0) as your test point unless the line goes through the origin
5. Substitute coordinates into the original inequality
6. Shade the side where your test point made the statement true

The Bottom Line

The half-plane you shade depends entirely on whether your test point satisfies the inequality. That's it. There's no shortcut, no trick, no special case where this fails.

Pick a point. Plug it in. Shade where it works.