Maths3 min readUpdated Apr 2, 2026

How to Graph Linear Inequalities

Master the step-by-step process of graphing linear inequalities, from boundary lines to shading regions, and learn how to visualize complex solution sets.

Key Takeaways

Strict inequalities (<, >) use dashed lines; inclusive inequalities (≤, ≥) use solid lines.
The shaded region represents every coordinate point (x, y) that makes the inequality true.
The origin (0,0) is typically the easiest test point to determine which side of the line to shade.
Always flip the inequality sign when multiplying or dividing by a negative number.
Systems of inequalities are solved by finding the overlapping "intersection" region.

Equations vs. Inequalities: The Solution Region

While an equation like y = 2x + 1 represents a single, infinitely thin line, a linear inequality like y > 2x + 1 represents an entire region of the coordinate plane. This shaded area contains an infinite number of points that satisfy the mathematical statement. Graphing an inequality allows you to visualize all possible solutions as a physical space rather than just a line. The border of this space is called the **boundary line**, and the side we shade is called the **solution set**.

The Boundary Line: Solid vs. Dashed

The first step is drawing the boundary line as if it were a normal equation. However, the type of line you draw communicates critical information about whether the points ON the line are part of the solution.

Symbol	Meaning	Line Type	Included?
<	Less than	Dashed	No
>	Greater than	Dashed	No
≤	Less than or equal	Solid	Yes
≥	Greater than or equal	Solid	Yes

Think of a dashed line like a "keep out" sign, it marks the edge but isn't part of the property. A solid line is like a fence that you’re allowed to touch.

Identifying the Shaded Region (Test Point Method)

Once the boundary line is drawn, you must decide which side to shade. The reliable standard is the **Test Point Method**. Pick any point not on the line, the origin (0,0) is almost always the easiest choice.

Step-by-Step

3 steps

Plug (0,0) into the inequality.

If the statement is TRUE, shade the side containing (0,0).

If the statement is FALSE, shade the opposite side.

If your boundary line goes exactly through (0,0), pick a different easy point like (1,0) or (0,1).

Graphing Using Slope-Intercept Shortcut

Before you can test points or shade, you need a precise boundary line. The most efficient way to draw this is by converting the inequality into Slope-Intercept form (y = mx + b).

Formula

y > 2x - 3

In this example, your y-intercept is -3 and your slope is 2/1 (up 2, right 1). Plot the intercept, use the slope to find the second point, and connect them with the appropriate line type (dashed for >).

Systems of Inequalities: Overlapping Regions

A system of inequalities consists of two or more inequalities graphed on the same coordinate plane. The final solution is not just any shaded area, but the specific region where the shading for every inequality overlaps. Technique for finding the overlap:

Graph the first inequality and shade lightly.

Graph the second inequality and shade lightly in a different direction.

The dark, double-shaded region is the final solution.

Points must satisfy ALL inequalities to be in this region.

Worked Example: Graphing y > 2x - 3

Let's follow the full workflow to graph y > 2x - 3:

Step-by-Step

6 steps

Identify the boundary: Draw y = 2x - 3.

Check the line type: Since the symbol is ">" (strict), use a dashed line.

Plot the points: Y-intercept at -3, next point at (1, -1).

Pick a test point: Use (0,0).

Test: 0 > 2(0) - 3 -> 0 > -3. This is TRUE.

Shade: Since the test was true, shade the side containing (0,0).

Frequently Asked Questions

When do I flip the inequality sign?

You must flip the symbol (e.g., change < to >) whenever you multiply or divide both sides of an inequality by a negative number. This is a common source of errors in algebra.

What does a shaded region actually represent?

The shaded region represents every single coordinate (x, y) that, when plugged into the inequality, results in a true statement. It is a visual representation of all possible answers.

Can I shade based on the inequality symbol alone?

If the inequality is in slope-intercept form (y > mx + b), ">" usually means "shade above" and "<" means "shade below." However, the Test Point Method is more reliable for complex or standard-form equations.

How do I graph inequalities with only one variable, like x > 4?

An inequality like x > 4 is a vertical boundary line at 4 on the x-axis. Since it is ">" (strict), use a dashed line and shade everything to the right.

What is a "System of Inequalities" used for?

Systems are used in real-world "Linear Programming" to find optimal solutions under constraints, such as maximizing profit within a limited budget and time.

How do I handle y ≤ 5?

This is a horizontal boundary line at 5 on the y-axis. Use a solid line (because of ≤) and shade everything below the line.

What if the test point (0,0) is on the boundary line?

If (0,0) is on the line, the test will be inconclusive. Choose a different point like (1,0) or (0,1) that is clearly on one side of the line.

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How to Graph Linear Inequalities

Key Takeaways

Equations vs. Inequalities: The Solution Region

The Boundary Line: Solid vs. Dashed

Identifying the Shaded Region (Test Point Method)

Graphing Using Slope-Intercept Shortcut

Systems of Inequalities: Overlapping Regions

Worked Example: Graphing y > 2x - 3

Frequently Asked Questions

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