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To understand the graph of an inequality like \(x + y < 4\), it's important to first visualize how it differs from the graph of an equation. When we graph an equation such as \(x + y = 4\), we get a straight line on the coordinate plane. However, an inequality like \(x + y < 4\) means that we are interested in the entire region where the sum of \(x\) and \(y\) is less than 4.

To sketch the graph, we first draw the line \(x + y = 4\). This is a boundary line that helps to divide the plane into two regions: one where \(x + y < 4\) and one where \(x + y > 4\). The line itself usually appears as a dashed line when graphing inequalities of "less than" or "greater than" to indicate that points on the line are not included in the solution set.

Next comes the shading part. Since (0,0) is verified as a solution point, we shade the region below the line \(x + y = 4\), covering all points (x, y) that satisfy \(x + y < 4\). This shaded area represents all the solutions to the inequality.

Determining if a particular point is a solution to an inequality involves substituting the point's coordinates into the inequality and checking if the inequality holds true. For example, for the inequality \(x + y < 4\), we substitute a point, say (0,0), into the inequality. We calculate to get \(0 + 0 \lt 4\), which is true. This means (0,0) lies within the region of solutions for this inequality.

Understanding "solution" in the context of inequalities involves recognizing that unlike equations, where solutions are often individual points, solutions to inequalities are sets or regions of points. So, after determining that a point like (0,0) is a solution, it provides a hint about which side of the boundary line encompasses all the solutions.

In general, to determine whether a point is in the solution set of an inequality:

• Substitute the point's coordinates into the inequality.
• If the inequality is satisfied, the point is within the solution region.
• If not, the point lies outside the solution region.

The coordinate plane is fundamental in graphing linear inequalities. It consists of two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes divide the plane into four quadrants, each containing points represented by ordered pairs (x, y).

When graphing an inequality like \(x + y < 4\), start by defining the boundary line using the corresponding equation (here \(x + y = 4\)). Find where this line intersects the axes to help plot it accurately, here at points (0,4) on the y-axis and (4,0) on the x-axis.

The coordinate plane is also where we use the determined solution points to help decide (by shading) which part of the plane contains all the solutions to the inequality. This methodology combines visual skills with mathematical calculation, making it easier to relate algebraic solutions to their graphical representations in two dimensions. This process unlocks a deeper understanding of how inequalities represent ranges or areas rather than single values.