Solving Inequalities with Graphs- Complete Tutorial

What You're Actually Learning Here

Solving inequalities with graphs sounds complicated until you realize it's just visual problem-solving. Instead of crunching numbers, you draw lines and shade regions. The answer sits right there on the coordinate plane.

Most students struggle because their textbooks overcomplicate this. Here's the truth: if you can draw a straight line and know which side of it to shade, you can solve these problems.

The Inequality Symbols You Need to Memorize

Before touching a graph, you need these symbols locked in your head:

• < = less than
• > = greater than
• ≤ = less than or equal to
• ≥ = greater than or equal to
• ≠ = not equal to (less common in graphing)

The solid line vs. dashed line rule comes from these symbols. Equal to variants (≤, ≥) get solid lines because the boundary is included. Strict inequalities (<, >) get dashed lines because the boundary is excluded.

Graphing a Single Inequality on a Number Line

This is the foundation. A number line graph shows where solutions exist along one dimension.

Steps for Number Line Graphs

Example: Graph x > 3

• Draw a number line with 3 marked
• Put an open circle at 3 (because 3 is NOT included)
• Shade everything to the right of 3
• Done

Example: Graph x ≤ -2

• Draw a number line with -2 marked
• Put a closed circle at -2 (because -2 IS included)
• Shade everything to the left of -2
• Done

Graphing Linear Inequalities in Two Variables

This is where coordinate planes come in. You're working with x and y instead of just x.

Example: Graph y ≤ 2x + 1

Step 1: Treat It Like an Equation First

Replace the inequality symbol with an equals sign: y = 2x + 1

This is your boundary line. Find two points, plot them, draw the line.

• When x = 0, y = 1 → point (0, 1)
• When x = 2, y = 5 → point (2, 5)
• Connect the dots

Step 2: Solid or Dashed?

Your symbol is ≤, so draw a solid line. The line itself is part of the solution.

Step 3: Pick a Test Point

Pick any point NOT on your line. The origin (0, 0) usually works.

Plug it into the original inequality: 0 ≤ 2(0) + 1

0 ≤ 1 is true. So shade the side containing (0, 0).

Quick Decision Table

Solving Systems of Inequalities Graphically

A system has multiple inequalities. The solution is where all shaded regions overlap.

Example: Solve the system

y > x - 2
y ≤ -x + 4

Step 1: Graph Each Inequality Separately

For y > x - 2:

• Draw y = x - 2 as a dashed line
• Test point (0, 0): 0 > -2 ✓
• Shade above the line

For y ≤ -x + 4:

• Draw y = -x + 4 as a solid line
• Test point (0, 0): 0 ≤ 4 ✓
• Shade below the line

Step 2: Find the Overlap

The solution is the region where both shadings exist. This is typically a wedge-shaped area bounded by your two lines.

Common Mistakes That Will Cost You Points

• Using solid lines for strict inequalities — if you see < or >, the line must be dashed
• Shading the wrong side — always test a point, especially when you're unsure
• Forgetting to solve for y — when graphing x > something, that's a vertical line, not horizontal
• Missing the overlap region — in systems, students often shade everything instead of finding the intersection

How to Solve an Inequality from a Graph

Sometimes you'll get a graph and need to write the inequality. Here's how:

Example

Given a solid line through (0, 3) and (2, 7), with the region below shaded:

• Find the slope: (7-3)/(2-0) = 2
• Find the y-intercept: 3
• Equation is y = 2x + 3
• Solid line means ≤ or ≥
• Below shading with (0,0) test gives 0 ≤ 3, which is true
• Answer: y ≤ 2x + 3

When to Use This Method

Graphing inequalities works best when:

• You have two variables and need to see the solution region
• You're solving optimization problems in linear programming
• You need to visualize constraints visually
• You're working with systems where the intersection matters

Algebra works fine for simple single-variable inequalities. But once you hit two variables, graphs make the problem actually manageable.