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To understand how to graph linear inequalities, such as y > 4x - 3, one must start by recognizing that this inequality is similar to the linear equation y = 4x - 3, but instead of a line, the inequality represents a half-plane, which is one side of the boundary line. The boundary line is dashed or solid depending on the type of inequality: a solid line for ≤ or ≥, and a dashed line for < or >. In this case, it will be dashed because the inequality strictly excludes values on the line.

After plotting the boundary line, the next step is to determine which side of the line to shade. Shading represents all the possible (x, y) coordinate solutions to the inequality. We achieve this through the test point method, which will be elaborated on in its respective section. It's important to note that this process is a visual representation of the solution set for the inequality.

The slope-intercept form of a line is y = mx + b, where m represents the slope, and b represents the y-intercept. This form is particularly useful for graphing because it gives you a starting point (b) on the y-axis and tells you how steeply the line is sloping and in which direction, based on the slope (m).

For our inequality, the slope-intercept form is y = 4x - 3. The y-intercept (b) is -3, meaning the line crosses the y-axis at (0, -3). The slope (m) is 4, or 4/1, indicating that for each step right on the x-axis, we move 4 steps up on the y-axis. This method gives you two points which allow you to draw the boundary line needed to solve the inequality.

The boundary line in the context of inequalities serves as the divider between the regions that satisfy the inequality and those that do not. It is based on the associated linear equation that comes from replacing the inequality sign with an equal sign. When graphing an inequality such as y > 4x - 3, the boundary line is the line y = 4x - 3.

For this inequality, the boundary line is dashed because it does not include the points where y is exactly 4 times x minus 3. The line itself is not part of the solution. To draw the boundary line, you simply plot the y-intercept at -3, and then use the slope to find another point. Such visual cues are vital for understanding which parts of the graph to consider for the solution set.

The test point method is a straightforward way to determine which side of a boundary line to shade when graphing a linear inequality. After drawing the boundary line, pick any point that does not lie on that line. The origin (0,0) is a common choice, unless the line passes through it.

In our inequality y > 4x - 3, we substitute (0,0) into the inequality to check if it holds. Since substituting the values gives us 0 > -3, which is a false statement, we do not shade the side containing (0,0), but rather the opposite side. The test point method assures us that we've shaded the correct region, and thus we have successfully graphed the solution to the inequality.