91Ó°ÊÓ

Let's start by understanding the slope-intercept form, the backbone of graphing linear equations and inequalities. This form is written as y = mx + b, where m represents the slope of the line and b is the y-intercept. The slope is the rate at which y changes with respect to x, effectively telling us how steep the line is. A positive slope means the line ascends from left to right, while a negative slope means it descends.

The y-intercept is the point where the line crosses the y-axis, indicated by the coordinate (0, b). It's here that we start plotting our graph. For our exercise, the inequality 2y - x < 8 is transformed into y < 0.5x + 4. Notice the slope, m, is 0.5, and the y-intercept, b, is 4. This tells us to begin at the point (0, 4) on the graph and, due to our slope of 0.5, for every 1 unit increase in y, we move 2 units to the right along the x-axis.

When graphing inequalities, we are not simply drawing a line but indicating an entire region of the coordinate plane that satisfies the inequality. Unlike equations, where one line contains all solutions, an inequality encompasses a vast array of points. In the step by step solution, after converting 2y - x < 8 to y < 0.5x + 4, the next phase is to graph the line y = 0.5x + 4.

However, because our inequality sign is '<' and not '=', this line is dashed to indicate that it is not included in the solution set; the points on this line don't satisfy y < 0.5x + 4. To graph this correctly, plot a dashed line through the points found with the slope and intercept. This dashed line is the boundary between what is included in the solution and what is not. It's akin to a fence on the graph - we are interested in what lies on one side of it.

After plotting the line that acts as the boundary as explained earlier, it's time to determine which side of the boundary contains the solutions to the inequality. This is where shading regions on a graph comes into play. Since our inequality uses '<', we look for the area where y is less than the line's y-values; we are interested in the lower region.

To be certain of which side to shade, we select a test point that is not on the line. The origin (0,0) is commonly used for this purpose. If the test point satisfies the inequality when substituted in, then its side is the solution region. For instance, substitution in our exercise gives us 0 < 4, confirming that the origin's side is correct. We then shade this entire side of the graph. This shaded area represents all the possible solutions to the inequality y < 0.5x + 4, meaning any point within this region makes the inequality true.