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The first step when working with a linear inequality like \(4x + 3y > 15\) is to convert it into the slope-intercept form. This form is essential for easy graphing and is expressed as \(y = mx + b\). Here, \(m\) stands for the slope of the line, and \(b\) represents the y-intercept, which is where the line crosses the y-axis.

To do this conversion, we first isolate 'y' in the inequality. Starting with the original inequality \(3y > -4x + 15\), we divide everything by 3, which yields \(y > -\frac{4}{3}x + 5\). This equation now tells us that the slope \(m\) is \(-\frac{4}{3}\), and the y-intercept \(b\) is 5.

Understanding the slope and intercept allows us to accurately plot the line on the graph, starting at the y-intercept (0,5) and using the slope to determine the direction and steepness of the line.

When graphing the inequality from our equation \(y > -\frac{4}{3}x + 5\), it's crucial to understand why we use a dashed line. In the context of inequalities, a dashed line indicates that the line is not included in the solution set. In other words, the inequality \(>\) or \(<\) suggests that the boundary line itself is not a part of the solutions.

To draw a dashed line, follow the usual steps of graphing a line using the slope and y-intercept, but make sure the line is composed of dashes rather than a solid stroke. This visually represents that points on the line don't satisfy the inequality; instead, solutions can be found either above or below this line, depending on the inequality.

After plotting the dashed line for the inequality \(y > -\frac{4}{3}x + 5\), the next step involves shading the correct area or region. Shading signifies all the possible solutions to the inequality.

Since the inequality is \(y > -\frac{4}{3}x + 5\), we need to shade the region above the dashed line because 'y' is greater than the expression \(-\frac{4}{3}x + 5\).

This means that any point in this shaded area will satisfy the inequality. It can be helpful to pick a test point, like the origin (0,0), to determine if it's part of the solution. If it satisfies the inequality, shade the area where that point lies; if not, shade the opposite area. In this case, the origin doesn't satisfy the inequality, confirming the shading above the line.