Which Is The Graph Of Linear Inequality 6x 2y 10

Which is the Graph of Linear Inequality 6x + 2y ≥ 10? A Comprehensive Guide

Understanding linear inequalities and their graphical representation is crucial in various fields, from mathematics and economics to computer science and engineering. This article delves deep into the linear inequality 6x + 2y ≥ 10, explaining how to graph it accurately and interpreting its meaning. We’ll explore the steps involved, address common misconceptions, and offer practical tips for solving similar problems.

Understanding Linear Inequalities

Before we tackle the specific inequality, let's establish a firm understanding of linear inequalities in general. A linear inequality is a mathematical statement that compares two linear expressions using inequality symbols such as:

• ≥: Greater than or equal to
• ≤: Less than or equal to
• >: Greater than
• <: Less than

Unlike linear equations, which represent a single line, linear inequalities represent a region on the coordinate plane. This region contains all the points (x, y) that satisfy the inequality.

Steps to Graph 6x + 2y ≥ 10

Graphing linear inequalities involves several key steps. Let's apply these steps to our example, 6x + 2y ≥ 10:

Step 1: Rewrite the Inequality as an Equation

To begin, we rewrite the inequality as an equation:

6x + 2y = 10

This equation represents the boundary line of the inequality's solution region.

Step 2: Find the x and y-intercepts

To easily plot the boundary line, we find its x and y-intercepts.

• x-intercept: Set y = 0 and solve for x: 6x + 2(0) = 10 6x = 10 x = 10/6 = 5/3 ≈ 1.67

• y-intercept: Set x = 0 and solve for y: 6(0) + 2y = 10 2y = 10 y = 5

Therefore, the line passes through the points (5/3, 0) and (0, 5).

Step 3: Plot the Boundary Line

Now, plot the points (5/3, 0) and (0, 5) on the Cartesian coordinate plane and draw a straight line connecting them. Crucially, because our original inequality is ≥, we draw a solid line. This indicates that the points on the line itself are included in the solution set. If the inequality were > or <, we would draw a dashed line to show that the points on the line are not included.

Step 4: Determine the Shaded Region

This is the most important step. To determine which side of the line represents the solution region, we choose a test point. The origin (0, 0) is usually the easiest to use, unless the line passes through the origin. Let's substitute (0, 0) into the original inequality:

6(0) + 2(0) ≥ 10 0 ≥ 10

This statement is false. Since the test point (0, 0) does not satisfy the inequality, the region containing (0, 0) is not part of the solution. Therefore, we shade the region on the opposite side of the line.

Step 5: The Completed Graph

The completed graph shows a solid line passing through (5/3, 0) and (0, 5), with the region above the line shaded. This shaded region represents all the points (x, y) that satisfy the inequality 6x + 2y ≥ 10.

Interpreting the Graph

The shaded region on the graph represents an infinite number of ordered pairs (x, y) that make the inequality 6x + 2y ≥ 10 true. Each point within this region satisfies the condition that the expression 6x + 2y is greater than or equal to 10.

Solving Similar Inequalities

The steps outlined above can be applied to any linear inequality. The key differences lie in:

• The inequality symbol: This determines whether the boundary line is solid (≥ or ≤) or dashed (> or <).
• The slope and intercepts: Different inequalities will have different lines.
• The shaded region: The test point will determine which side of the line to shade.

Common Mistakes to Avoid

Several common mistakes can lead to inaccurate graphs:

• Incorrect line type: Using a dashed line when a solid line is required (or vice versa).
• Shading the wrong region: Choosing an inappropriate test point or misinterpreting the result.
• Mathematical errors: Incorrectly calculating the intercepts or solving the inequality.
• Improper scaling: Not properly scaling the axes can distort the graph and lead to inaccurate conclusions.

Advanced Concepts and Applications

The graphing of linear inequalities forms the basis for understanding more advanced concepts such as:

• Systems of linear inequalities: These involve multiple inequalities, and the solution region is the intersection of the individual solution regions.
• Linear programming: This optimization technique uses linear inequalities to model constraints and find optimal solutions.
• Feasible regions: In optimization problems, the solution region of a system of linear inequalities is often referred to as the feasible region, representing all possible solutions that satisfy the constraints.

Practical Applications

Linear inequalities find applications in various real-world scenarios:

• Resource allocation: Businesses use linear inequalities to optimize resource allocation, ensuring that production meets demand while staying within budget.
• Production planning: Manufacturing companies use linear programming techniques based on linear inequalities to plan production schedules effectively.
• Financial modeling: Investors use linear inequalities to model financial constraints and risk tolerance.
• Game theory: Linear inequalities play a role in analyzing strategic interactions and decision-making in games.

Conclusion

Graphing the linear inequality 6x + 2y ≥ 10 involves a methodical approach: rewriting the inequality as an equation, finding intercepts to plot the boundary line, choosing a test point to determine the shaded region, and then interpreting the resulting graph. Understanding these steps is fundamental for solving similar problems and applying linear inequalities to various real-world contexts. By mastering these techniques, you can confidently tackle more complex problems involving systems of inequalities and optimization techniques, opening doors to further exploration in mathematics and its applications. Remember to always double-check your calculations and carefully consider the implications of the inequality symbols to avoid common errors and ensure accurate graphical representation.