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Understanding the boundary line is essential when graphing linear inequalities. The boundary line represents all the points that satisfy the linear equation that is associated with the inequality. In our example, we convert the inequality \(5x + 4y > 4\) into the equation \(5x + 4y = 4\).

This boundary line is crucial as it divides the entire coordinate plane into two distinct regions. One of those regions will contain the solution set to the inequality, while the other will not. The type of line drawn depends on the inequality sign. For strict inequalities (like \(>\) or \(<\text{\) the boundary line is dashed, indicating that points on this line are not part of the solution. Conversely, for inclusive inequalities (\(≥\) or \(≤\)), the line is solid to demonstrate that points on the line are included in the solution set. Recognizing this distinction is crucial for accurately graphing linear inequalities.

Intercepts are points where the graph of an equation crosses the axes. The x-intercept is the point where the graph intersects the x-axis, and the y-intercept where it intersects the y-axis. To find the x-intercept, we substitute \(y = 0\) into our equation and solve for \(x\), which gives us the point on the x-axis that the line goes through. Similarly, for the y-intercept, we set \(x = 0\) in our equation and solve for \(y\).

These intercepts are the key to sketching the boundary line since they provide two points through which the line passes. For the equation \(5x + 4y = 4\), finding the intercepts allows us to plot two definite points, making it easier to draw the boundary line with precision. This step simplifies the graphing process and serves as a foundation for further analysis of the inequality’s solution.

Shading is used to indicate the solution set of an inequality on a graph. Once the boundary line is drawn, we need to determine which side of the line represents the solution. We can do this by selecting a test point not on the boundary line, often the origin \((0,0)\), and substituting it into the original inequality.

If the test point satisfies the inequality, then the region of the graph containing that point is shaded. Otherwise, we shade the opposite side. For the inequality \(5x + 4y > 4\), if \((0,0)\) does not satisfy the inequality, as it doesn't in this case, we then shade the region above the line since \(y\) is positive, and the inequality symbol is \(>\). Shading effectively visualizes the set of all points that are solutions to the inequality, which is one of the ultimate goals when graphing linear inequalities.