Graphing Inequality Regions on Graph Paper: A Step-by-Step Guide

Graphing inequalities can be a challenging process, but with a systematic approach, it becomes much more manageable. This tutorial will guide you through the steps to draw the regions of the given inequalities on graph paper, specifically the inequalities 2x 3 ≥ 3y - x - 3 and 2y ≥ 4 - 2x. By the end of this piece, you will understand how to convert these inequalities into their equality forms, graph the boundary lines, and determine the regions that satisfy the inequalities.

Step 1: Convert Inequalities to Equality Forms

To start, we need to rewrite the given inequalities in their equality forms:

INEQUALITY 1:
2x 3 ≥ 3y - x - 3

First, simplify the inequality:

2x 3 ≥ 3y - x - 3

Add x and 3 to both sides:

2x x 3 3 ≥ 3y

3x 6 ≥ 3y

Divide both sides by 3:

x 2 ≥ y

or

y ≤ x 2 …….1

INEQUALITY 2:
2y ≥ 4 - 2x

Subtract 2x from both sides:

2y 2x ≥ 4

Simplify:

y x ≥ 2

or

y ≥ -2x 2 …….2

Step 2: Graph the Boundary Lines

Now, let's graph the associated equality forms to form the boundary lines:

EQUATION 1:
y x 2

Plot the line through the points (0, 2) and (-2, 0).

EQUATION 2:
y -2x 2

Plot the line through the points (0, 2) and (1, 0).

Step 3: Determine the Shadowed Region

The next step is to determine the regions that satisfy the inequalities. Since both inequalities have an “≥” (greater than or equal to) sign, the shaded region includes the boundary lines. Therefore, we need to shade the region below both lines.

To achieve this, follow these steps:

Graph the line y x 2. This line passes through the points (0, 2) and (-2, 0). Graph the line y -2x 2. This line passes through the points (0, 2) and (1, 0). Identify the intersection point of the lines. The intersection point of y x 2 and y -2x 2 can be found by solving the system of equations:

Setting y x 2 equal to y -2x 2:

x 2 -2x 2

3x 0

x 0

Substitute x 0 into either equation:

y 0 2 2

Thus, the intersection point is (0, 2).

The region we are interested in is below both lines, which includes the lines themselves since the inequalities are “≥”. Therefore, shade the region below the lines y x 2 and y -2x 2.

Conclusion

By following these steps, you can accurately graph the region that satisfies the given inequalities. This process can be applied to other inequalities as well. Remember to always graph the boundary lines first and then determine the correct region based on the “≥” or “≤” signs. This method ensures that you cover all the necessary steps and have a clear understanding of the solution.