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The boundary line is an essential concept when graphing inequalities, as it helps visually represent the equation on a graph. In our exercise, the inequality is initially given as \( 2y - x \ge 4 \). To find the boundary line, start by converting the inequality into an equation by replacing the inequality sign with an equal sign, obtaining \( y = 0.5x + 2 \). This line acts as a dividing line on the graph between the solution area and the non-solution area.

When plotting the boundary line on a graph, it is crucial to consider whether the line is solid or dashed. In this case, since the inequality uses \(\ge \) (greater than or equal to), a solid line is used to include points that lie exactly on the line as part of the solution set.

The y-intercept is the point where the line crosses the y-axis, indicating where the value of x is zero. To find the y-intercept of the boundary line equation \( y = 0.5x + 2 \), observe the constant term, which is 2. This means the y-intercept is located at the point \( (0,2) \).

Graphically, you begin by plotting the y-intercept on the coordinate plane, establishing a starting point for the boundary line. From the y-intercept, the rest of the line can be drawn using the slope. Understanding the y-intercept is crucial as it provides the initial point from which the slope will extend the boundary line through the graph.

The slope of a line indicates its steepness and direction. It is defined by the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. For the boundary line \( y = 0.5x + 2 \), the slope is 0.5, meaning for every unit increase in x, y increases by 0.5 units.

To graph the line using the slope, start at the y-intercept \( (0,2) \). From this point, move right 1 unit (run) and up 0.5 units (rise) to find the next point \( (1,2.5) \). Draw a straight line through these points to visualize the boundary line. Slope is a pivotal component as it not only helps in plotting the line but also helps understand the relationship between dependent and independent variables.

Shading a region is the final step in graphing inequalities and demonstrates where the solutions to the inequality lie on the graph. Once the boundary line, \( y = 0.5x + 2 \), is drawn, determine which side of the line to shade by considering the inequality sign. Since the inequality is \( y \ge 0.5x + 2 \), it indicates that the solution includes all y-values that are greater than or equal to those on the line, meaning you shade above the line.

The shaded region represents all the coordinate pairs \( (x, y) \) that satisfy the inequality. By shading the correct region, you effectively depict all potential solutions to the inequality, allowing it to be easily visualized by anyone viewing the graph. This graphical representation is crucial in interpreting and solving inequalities through visual means.