How Do You Graph Inequalities?

Graphing inequalities involves visually representing all possible solutions on a coordinate plane. The basic process involves treating the inequality like an equation, graphing it, and then shading the appropriate region.

Here's a breakdown of the steps:

1. Treat the Inequality as an Equation: Replace the inequality symbol (<, ≤, >, or ≥) with an equals sign (=). For example, if you have y > 2x + 1, pretend it's y = 2x + 1.

2. Graph the Equation: Graph the resulting equation. This will be a line (linear inequality) or another curve depending on the equation.

3. Determine the Line Type (Solid or Dashed): This is crucial.

   • If the inequality is strict (< or >), use a dashed (or dotted) line. This indicates that the points on the line are not included in the solution.
   • If the inequality is inclusive (≤ or ≥), use a solid line. This indicates that the points on the line are included in the solution.

4. Choose a Test Point: Select a point that is not on the line you just graphed. The easiest point is usually (0,0), unless the line goes through the origin.

5. Substitute the Test Point into the Original Inequality: Plug the x and y coordinates of your test point into the original inequality.

6. Determine Shading:

   • If the inequality is true when you substitute the test point, shade the side of the line that contains the test point. This area represents all the solutions to the inequality.
   • If the inequality is false when you substitute the test point, shade the side of the line that does not contain the test point.

Example:

Let's graph the inequality y ≤ x + 2.

1. Equation: y = x + 2

2. Graph: Graph the line y = x + 2. It has a slope of 1 and a y-intercept of 2.

3. Line Type: Since the inequality is ≤ (less than or equal to), use a solid line.

4. Test Point: Choose (0,0).

5. Substitution: 0 ≤ 0 + 2 which simplifies to 0 ≤ 2.

6. Shading: Since 0 ≤ 2 is true, shade the side of the line that contains (0,0). This is the area below the line.

In Summary: Graphing inequalities involves graphing a related equation and then shading the region that contains all points that satisfy the inequality. Remember to pay attention to whether the boundary line should be solid or dashed.