Graph Linear Inequalities- Algebraic Visualization Techniques

What Linear Inequalities Actually Are

Linear inequalities are just equations with a twist. Instead of an equals sign, you get <, >, ≤, or ≥. They represent a range of solutions, not a single line.

When you graph them, you don't get one clean line. You get a region — a chunk of the coordinate plane where every point satisfies the inequality.

That's the whole concept. Everything else is technique.

The Boundary Line: Your Starting Point

Every linear inequality has a boundary line. Find it by replacing the inequality sign with an equals sign.

For y ≤ 2x + 3, the boundary line is y = 2x + 3.

Graph that line first. Then decide what kind of line you need:

• Solid line — use when the inequality includes ≤ or ≥ (the boundary is part of the solution)
• Dashed line — use when the inequality is strict: < or > (the boundary is NOT part of the solution)

The Test Point Method: How to Actually Shade It

Once you have your boundary line, you need to shade the correct region. The test point method works every time.

Step-by-Step Process

1. Graph the boundary line (solid or dashed based on your inequality)
2. Pick a test point not on the line. Use (0, 0) if it's not on the line — it's the easiest
3. Substitute the test point into the original inequality
4. If the test point makes a true statement, shade the region containing it
5. If it's false, shade the opposite region

Quick Example

Graph y > x - 2

1. Boundary line: y = x - 2. Use a dashed line because it's strict (>)
2. Test point: (0, 0)
3. Plug in: 0 > 0 - 2 → 0 > -2 → TRUE
4. Shade the region containing (0, 0)

Done.

Special Cases That Trip People Up

When the Test Point Is On the Line

Can't use (0, 0) if it sits on your boundary line. Pick another point like (1, 0) or (0, 1). Any point not on the line works.

Horizontal and Vertical Boundary Lines

For y ≤ -4, the boundary is a horizontal line at y = -4. You still graph it, pick a test point above or below, and shade accordingly.

Same process for x > 2 — vertical boundary line, test left or right.

Standard Form Inequalities

For Ax + By ≤ C, solve for y first if B is negative. You want y alone on one side so the inequality direction stays correct when dividing by negative numbers.

Example: 2x + 3y ≤ 6

1. Subtract 2x: 3y ≤ -2x + 6
2. Divide by 3: y ≤ (-2/3)x + 2
3. Now graph y = (-2/3)x + 2 with a solid line
4. Test (0, 0): 0 ≤ 2 → TRUE, shade toward the origin

Inequality Symbols: A Quick Reference

Note: "Above" and "below" depend on the slope. The test point method always tells you the correct side.

How to Graph Linear Inequalities: Quick Start Guide

1. Isolate y if needed (get it by itself on the left)
2. Graph the boundary line using y = mx + b form
   • Solid if ≤ or ≥
   • Dashed if < or >
3. Pick a test point not on the line (try origin first)
4. Evaluate the inequality at that point
5. Shade the half-plane containing your true test point

Common Mistakes That Produce Wrong Graphs

• Using a solid line for strict inequalities — (< or >) always need dashed lines
• Forgetting to flip the inequality sign when dividing by a negative number
• Shading the wrong side — always verify with a test point
• Drawing the line wrong — check your y-intercept and slope before worrying about shading

Putting It Together: Full Example

Graph the inequality: 3y - 6 > 2x

Step 1: Put in slope-intercept form

3y > 2x + 6

y > (2/3)x + 2

Step 2: Graph y = (2/3)x + 2 as a dashed line

Step 3: Test point (0, 0)

0 > (2/3)(0) + 2

0 > 2

FALSE

Step 4: Shade the region NOT containing (0, 0)

The correct region is above and to the right of the line.

Why This Skill Matters

Linear inequalities appear in optimization problems, systems of inequalities, and real-world constraint problems. If you can't graph them correctly, you'll struggle with everything that follows.

The test point method works for every linear inequality. Memorize it. Use it every time. Don't guess the shading direction.