How to Graph an Inequality?

Graphing an inequality involves visualizing the solution set on a coordinate plane. Here's a step-by-step guide:

1. Replace the Inequality Symbol with an Equal Sign: Treat the inequality as a regular equation for now. This equation represents the boundary line of the solution region.

2. Graph the Equation: Graph the line corresponding to the equation you created in step 1.

   • If the inequality is strict (using "<" or ">"), draw a dashed line. This indicates that the points on the line are not included in the solution.
   • If the inequality includes "or equal to" (using "≤" or "≥"), draw a solid line. This indicates that the points on the line are included in the solution.

3. Choose a Test Point: Select a point not on the line you just graphed. The point (0, 0) is often the easiest to use if the line doesn't pass through the origin.

4. Test the Point in the Original Inequality: Substitute the coordinates of your test point into the original inequality.

5. Determine the Shaded Region:

   • If the test point satisfies the inequality (makes the inequality true), shade the region of the graph that contains the test point. All points in this region are solutions to the inequality.
   • If the test point does not satisfy the inequality (makes the inequality false), shade the region of the graph that does not contain the test point.

Example:

Let's graph the inequality: y > 2x + 1

1. Replace with Equal Sign: y = 2x + 1

2. Graph the Line: This is a line with a slope of 2 and a y-intercept of 1. Because the original inequality uses ">", we draw a dashed line.

3. Choose a Test Point: Let's use (0, 0).

4. Test the Point: Substitute (0, 0) into y > 2x + 1: 0 > 2(0) + 1 which simplifies to 0 > 1.

5. Determine the Shaded Region: 0 > 1 is false. Therefore, we shade the region above the dashed line (the region that does not contain (0, 0)).

Summary Table:

Inequality Symbol	Line Type	Shaded Region
>	Dashed	Above the line (for y >) or Right of the line (for x >)
<	Dashed	Below the line (for y <) or Left of the line (for x <)
≥	Solid	Above the line (for y ≥) or Right of the line (for x ≥)
≤	Solid	Below the line (for y ≤) or Left of the line (for x ≤)