MathsFrancais3 min de lectureMis a jour 2 avr. 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.

Points Cles

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. ``` 1. Plug (0,0) into the inequality. 2. If the statement is TRUE, shade the side containing (0,0). 3. 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). ``` 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: ``` 1. Identify the boundary: Draw y = 2x - 3. 2. Check the line type: Since the symbol is ">" (strict), use a dashed line. 3. Plot the points: Y-intercept at -3, next point at (1, -1). 4. Pick a test point: Use (0,0). 5. Test: 0 > 2(0) - 3 -> 0 > -3. This is TRUE. 6. Shade: Since the test was true, shade the side containing (0,0). ```