Graphing Linear Inequalities: Techniques and Examples

Graphing linear inequalities is a fundamental skill in algebra, helping to visualize the solution set of an inequality in a geometric context. This article will guide you through the process of graphing linear inequalities in both the coordinate plane and on a number line, providing detailed steps and examples for clarity.

Graphing Linear Inequalities in the Coordinate Plane

Graphing a linear inequality in the coordinate plane involves several steps. This section will walk you through the process, ensuring each step is explained clearly.

Steps to Graph a Linear Inequality

1. Write the Inequality in Standard Form: Ensure your inequality is in the form A x B y ≤ C or A x B y ≥ C.

2. Graph the Boundary Line: Replace the inequality sign with an equals sign to find the boundary line. For example, if your inequality is y ≥ 2x - 3, graph the line y 2x - 3. Determine whether to use a solid or dashed line: Solid Line: If the inequality is ≤ or ≥, the boundary includes the points on the line. Dashed Line: If the inequality is or , the boundary does not include the points on the line.

3. Find Points: Find two or more points on the line to plot. You can use the intercepts or substitute values for x to find corresponding y values.

4. Shade the Appropriate Region: Choose a test point that is not on the boundary line. The origin (0, 0) is often a convenient choice unless it lies on the line. Substitute the test point into the original inequality: If the inequality holds true, shade the region that includes the test point. If the inequality does not hold true, shade the opposite side.

Example:

Graph the inequality y ≥ 2x - 1. Boundary Line: The boundary line is y 2x - 1. Graph the Line: Plot the line using a dashed line since it's a strict inequality ≥. Test Point: Use the point (0, 0): Substitute: 0 ≥ 2(0) - 1 → 0 ≥ -1 is true. Shade the region that includes (0, 0). Final Graph: You will have a dashed line representing y 2x - 1 with the area below the line shaded, indicating all the points where y ≥ 2x - 1.

Graphing Linear Inequalities on a Number Line

Graphing linear inequalities on a number line involves representing the solution set as a set of numbers on a number line. This section will provide a step-by-step guide to graphing linear inequalities on a number line.

1. Write the inequality in standard form: For example, if the inequality is 2x - 3 ≤ 5, subtract 3 from both sides to get 2x ≤ 8 and then divide both sides by 2 to get x ≤ 4.

2. Plot the Point Corresponding to the Solution on the Number Line: For the inequality x ≤ 4, this point is 4.

3. Draw an Open Dot at the Solution Point: This indicates that the solution point is not included in the solution set. For the inequality x ≤ 4, draw an open dot at 4 on the number line.

4. Draw an Arrow Pointing to the Appropriate Side: Draw an arrow pointing to the left if the inequality is less than or equal to (≤) and to the right if the inequality is greater than or equal to (≥). For the inequality x ≤ 4, draw an arrow pointing to the left.

5. Shade the Region on the Number Line: For the inequality x ≤ 4, shade the region to the left of 4.

6. Label the Solution Set: Write the inequality symbol and the solution. For the inequality x ≤ 4, label the solution set as x ≤ 4.

This method can be applied to any linear inequality of the form ax b ≤ c or ax b ≥ c.