Graphing the Linear Inequality 2xy ≥ 6

When working with linear inequalities like 2xy ≥ 6, one of the crucial steps is to understand how to shade the correct region on the graph. This involves not just plotting the boundary line but also determining which side of the line to shade. Let's explore the step-by-step process in detail.

1. Understanding the Inequality

The inequality 2xy ≥ 6 can be thought of as the graphical representation of all points (x, y) that satisfy the condition that the value of 2x y is greater than or equal to 6. This can be visualized as the area above the line 2x y 6 on the coordinate plane, including the points on the line itself.

2. Steps to Graph the Inequality

To graph the linear inequality 2xy ≥ 6, follow these steps:

Step 1: Find the Boundary Line

To find the boundary line, start by rewriting the inequality in terms of y. For 2xy ≥ 6, isolate y on one side:

y ≥ 6 - 2x

The boundary line is the equation y 6 - 2x. This line divides the coordinate plane into two regions: one above the line and one below the line.

Step 2: Plot the Boundary Line

Choose two points that satisfy the equation y 6 - 2x to plot the line. For instance:

When x 0, y 6 - 2(0) 6, so the point (0, 6) is on the line. When x 3, y 6 - 2(3) 0, so the point (3, 0) is on the line.

Plot these points and draw a straight line connecting them. This line is the boundary for our inequality.

Step 3: Determine the Shaded Region

To determine which region to shade, choose a test point. A common test point is the origin (0, 0). Substitute (0, 0) into the original inequality:

2(0)(0) ≥ 6

This simplifies to 0 ≥ 6, which is false. Therefore, the point (0, 0) is not part of the solution. This means the region above the line is the solution to the inequality 2xy ≥ 6.

Graphing the Linear Equation and Inequality

Let's reiterate the key steps:

Equate the inequality to the boundary line: y 6 - 2x. Plot the boundary line using two points: (0, 6) and (3, 0). Test a point, such as (0, 0), to determine the correct region to shade. Since (0, 0) does not satisfy 2xy ≥ 6, shade the region above the line.

By following these steps, you can accurately graph and represent the inequality 2xy ≥ 6 on the coordinate plane.

Conclusion

Graphing linear inequalities is a fundamental skill in algebra and is particularly useful in understanding the behavior of various functions. By mastering these steps, you can easily visualize and solve a wide range of inequality problems.