graph the linear inequality shown below on the provided graph.

Guri Bee

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22-10-2024

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Mastering Linear Inequalities: A Visual Guide with Graphing

Linear inequalities, like their equation counterparts, represent relationships between variables. However, unlike equations, inequalities express a range of possible solutions instead of a single point. This makes them crucial in understanding real-world scenarios where constraints and limitations exist.

This article will walk you through the process of graphing linear inequalities. We will use examples and explanations based on real-world applications to help you visualize and understand the concepts.

Understanding the Basics

A linear inequality is an expression that involves variables, constants, and comparison symbols (<, >, ≤, ≥). The inequality symbol determines the region on the graph that represents the solution set.

Graphing Linear Inequalities: A Step-by-Step Guide

Let's consider the inequality:

• y > 2x - 1

Step 1: Graph the Boundary Line

• Treat the inequality as an equation: y = 2x - 1.
• Find the x and y intercepts (where the line crosses the axes) for this equation:
  • When x = 0, y = -1. Plot (0, -1) on the graph.
  • When y = 0, 0 = 2x - 1, so x = 1/2. Plot (1/2, 0) on the graph.
• Draw a straight line connecting these two points. This line represents the boundary of the solution set.

Step 2: Determine the Shaded Region

• Choose a test point: Select a point that is not on the boundary line. (0, 0) is often a convenient choice.
• Substitute the test point into the inequality:
  • 0 > 2(0) - 1, which simplifies to 0 > -1.
• Interpret the result: Since the inequality is true, the test point (0, 0) lies in the solution region.

Step 3: Shade the Solution Region

• Since the test point (0, 0) is in the solution region, shade the side of the boundary line that contains this point.

Important Note:

• If the inequality is strict (>, <), the boundary line is dashed to indicate that points on the line are not part of the solution.
• If the inequality is non-strict (≥, ≤), the boundary line is solid to indicate that points on the line are part of the solution.

Visualizing the Solution

The shaded region in the graph represents all the possible (x, y) pairs that satisfy the inequality y > 2x - 1.

Real-World Application: Budgeting

Imagine you have a budget of $100 for groceries. You plan to buy apples ($2 each) and oranges ($1 each). This scenario can be represented by the inequality:

• 2x + y ≤ 100 (where x represents the number of apples and y represents the number of oranges).

Graphing this inequality would show all the possible combinations of apples and oranges you could purchase within your budget.

Conclusion

Graphing linear inequalities provides a visual representation of the infinite solutions that satisfy the given conditions. This process is essential in visualizing constraints and understanding the relationship between variables in real-world applications. By mastering these steps, you can unlock a powerful tool for analyzing and solving problems that involve inequalities.