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Understanding the boundary line is essential when graphing inequalities. The line represents all the points that satisfy the equation when the inequality sign is replaced with an equals sign. For example, in the inequality \(y \leq x-3\), the boundary line is found by graphing \(y = x - 3\).

To represent this on a graph, you would plot points where the equation holds true. In our case, when \(x = 0\), \(y = -3\), and when \(x = 3\), \(y = 0\). These points give you a visual guide to draw the boundary line. It is crucial to use a solid line for inequalities including \(\leq\) or \(\geq\), as this indicates that points on the line are included in the solution set. If the inequality were \(<\) or \(>\), a dotted line would be used instead, to show that points on the line are not part of the solution.

After establishing the boundary line on a graph, you need to identify the region that contains all the solutions to the inequality. This is where shading comes into play. With the inequality \(y \leq x - 3\), we understand that the solution includes points where \(y\) is less than or equal to \(x - 3\).

To convey this visually, shade the region where the inequality holds true. Using the boundary line as a divider, one side will represent the solutions, while the other will not. For \(y \leq x - 3\), since the point \((0,0)\) fails to satisfy the inequality, it informs us that the region containing \((0,0)\) is not the solution. Consequently, the area below the solid boundary line is shaded, communicating effectively that every point in that shaded region satisfies the inequality.

The final step in graphing an inequality is quite often confirming the region to shade. This is done with a test point. The test point is a tool used to determine which side of the boundary line contains the solutions to the inequality.

Choose a point that is not on the boundary line, typically \((0,0)\) is used for simplicity, unless it lies on the line. Substitute this test point into the original inequality. If the inequality is true with the test point values, it means all points on that side of the line are included in the solution region. For our example, when plugging \((0,0)\) into \(y \leq x - 3\), we get \(0 \leq 0 - 3\), a false statement. This tells us that the opposite side of the boundary line from \((0,0)\) is the solution set, hence why we shade below the line. Test points provide a simple check to ensure that we are shading the correct region for the given inequality.