Graphing Linear Inequalities: A Step-by-Step Guide

Graphing Linear Inequalities: A Step-by-Step Guide

Linear inequalities are mathematical expressions that compare two quantities using inequality symbols such as <, >, ≤, or ≥. Graphing linear inequalities helps visualize the solutions to these inequalities, which represent a range of values that satisfy the given condition.

Understanding the Basics

Before we delve into graphing, let’s review the fundamental concepts:

• Inequality Symbols:
• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)
• Boundary Line: The line that represents the equality part of the inequality. It is the line that separates the solutions from the non-solutions.
• Shading: The region on the graph that represents all the points that satisfy the inequality.

Steps to Graphing Linear Inequalities

Follow these steps to graph a linear inequality:

1. Rewrite the inequality in slope-intercept form (y = mx + b): This form makes it easier to identify the slope (m) and y-intercept (b) of the boundary line.
2. Graph the boundary line:
• If the inequality includes < or >, draw a dashed line to indicate that the points on the line are not included in the solution.
• If the inequality includes ≤ or ≥, draw a solid line to indicate that the points on the line are included in the solution.
3. Choose a test point: Select any point that is not on the boundary line. It’s often easiest to choose (0, 0) if it’s not on the line.
4. Substitute the test point into the original inequality:
• If the inequality is true, shade the region containing the test point.
• If the inequality is false, shade the region that does not contain the test point.

Example: Graphing y < 2x + 1

Let’s graph the inequality y < 2x + 1:

1. Slope-intercept form: The inequality is already in slope-intercept form (y = mx + b), where m = 2 (slope) and b = 1 (y-intercept).
2. Graph the boundary line: Since the inequality uses <, draw a dashed line. The y-intercept is 1, so plot the point (0, 1). The slope is 2, meaning for every 1 unit increase in x, y increases by 2 units. Plot another point using the slope, such as (1, 3), and draw a dashed line through both points.
3. Choose a test point: We can choose (0, 0) as our test point.
4. Substitute and test: Substitute (0, 0) into the original inequality: 0 < 2(0) + 1. This simplifies to 0 < 1, which is true. Therefore, shade the region containing the test point (0, 0).

The graph of y < 2x + 1 will show a dashed line with the region below the line shaded.

Key Points to Remember

• The boundary line is dashed for < or > inequalities and solid for ≤ or ≥ inequalities.
• The shaded region represents all the points that satisfy the inequality.
• Always test a point to determine which side of the boundary line to shade.

By following these steps, you can confidently graph linear inequalities and visualize their solutions. Practice with various examples, and remember to always check your work by substituting a test point into the original inequality.