9th grade math - Algebra

Graphing Linear Inequalities

Visualizing Inequalities in Two Variables

You’ve already learned how to graph linear equations and understand the meaning of inequalities like x > 3 or y ≤ -2. Now, we combine those ideas to graph linear inequalities—inequalities that involve two variables, usually x and y, and describe a whole region of the coordinate plane, not just a line.

This article builds directly on what you’ve learned in previous lessons: how to graph linear equations, work with inequalities, and understand slope and y-intercepts. If you’re comfortable with those topics, you’re ready for graphing linear inequalities.

What Is a Linear Inequality?

A linear inequality is like a linear equation, but instead of an equals sign (=), it uses an inequality symbol:

  • < (less than)
  • ≤ (less than or equal to)
  • > (greater than)
  • ≥ (greater than or equal to)

Examples:

  • y < 2x + 3
  • y ≥ -x + 1

These describe a set of points, not just one line. The solution to a linear inequality is all the coordinate points (x, y) that make the inequality true.

Step-by-Step: How to Graph a Linear Inequality

1. Start by Graphing the Boundary Line

Ignore the inequality symbol for a moment and graph the related linear equation. For example, if the inequality is:

y < 2x + 1

First, graph the line y = 2x + 1. Use the slope (2) and the y-intercept (1) to draw the line.

2. Decide Whether the Line Is Solid or Dashed

  • Use a solid line if the inequality includes “equal to” (≤ or ≥). That means points on the line are included.
  • Use a dashed line if the inequality is just < or >. That means points on the line are not included.

In our example, since we have <, we’ll draw a dashed line.

3. Shade the Correct Side of the Line

This step is what makes inequalities different from equations.

After you draw the line, you need to shade the region that includes all the solutions.

To decide which side to shade:

  • Pick a test point not on the line (usually (0, 0) if possible).
  • Plug it into the inequality.

Example: Use y < 2x + 1 with test point (0, 0):

0 < 2(0) + 1 → 0 < 1 → true

Since the test point works, shade the side of the line that includes (0, 0).

Example: Graph y ≥ -x + 4

  1. Graph the line y = -x + 4. Slope = -1, y-intercept = 4
  2. Since the symbol is ≥, draw a solid line
  3. Test (0, 0): 0 ≥ -0 + 4 → 0 ≥ 4 → false
  4. Shade the side not containing (0, 0)

Special Cases

Vertical Lines

Inequality: x > 2

  • Vertical dashed line at x = 2
  • Shade to the right

Horizontal Lines

Inequality: y ≤ -1

  • Horizontal solid line at y = -1
  • Shade below the line

Real-World Connection

Graphing linear inequalities is useful in situations where you’re working with constraints or limits. For example:

  • A business must stay under a budget (≤)
  • A vehicle must go faster than a minimum speed (>)
  • Time spent on activities must not exceed a total limit (≤)

These situations don’t give one answer—they give a range of possibilities. Graphing helps you see that range visually.

How This Builds on What You Know

You already know how to:

Now, you’re using all that knowledge to graph regions defined by inequalities.

Practice Challenge

Try graphing this inequality on your own:

y < -2x + 5
  • Step 1: Graph y = -2x + 5
  • Step 2: Use a dashed line
  • Step 3: Test (0, 0): 0 < -2(0) + 5 → true
  • Step 4: Shade the side that includes (0, 0)

What’s Next?

Next up is graphing systems of linear inequalities—where you combine two or more inequalities to find a solution region that satisfies all of them at the same time!

Keep practicing—your graphing skills are leveling up fast!