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The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The intersection of these two axes at the point labeled (0, 0), is called the origin.

In graphing, we use the coordinate plane to plot points, lines, and inequalities. Each point on the plane is represented by a pair of numbers : (x, y), where 'x' denotes the position along the horizontal axis, and 'y' denotes the position along the vertical axis. The entire plane is divided into four quadrants, with positive and negative values for x and y controlling the quadrant in which a point is located.

Understanding the coordinate plane is crucial as it allows us to visually interpret algebraic equations and inequalities, as well as to clearly communicate spatial information.

The slope-intercept form is an equation of a line in the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. The slope indicates the steepness of the line and its direction, with positive values for upward slant and negative values for a downward slant from left to right.

For example, to graph the inequality \(\frac{2}{3} x+2 \leq \frac{2}{9} y\), we first convert it to slope-intercept form. By manipulating the inequality into \(y \geq 3x + 9\), we have identified the slope, \(m = 3\), and the y-intercept, \(b = 9\). This form allows us to easily plot the y-intercept on the y-axis and use the slope to find additional points to draw the line.

A boundary line in the context of graphing linear inequalities represents the edge of the region where the inequality is true or false. It is the graphical representation of the linear equation derived from the inequality by replacing the inequality sign with an equal sign.

Depending on the inequality sign, the boundary line can be solid or dashed. A solid line, as used when graphing \(y \geq 3x + 9\), indicates the points on the line are solutions to the inequality (\(\leq\) or \(\geq\)). A dashed line would show that points on the line are not part of the solution (\(<\) or \(>\)).

Graphing the boundary line is a stepwise process: start at the y-intercept on the y-axis, then follow the slope to mark another point, and finally draw the line through the points.

Shading is the process of marking the region of the coordinate plane that satisfies a given inequality. When graphing the solution set of an inequality, such as \(y \geq 3x + 9\), shading helps visually communicate which side of the boundary line contains solutions.

In this case, because the inequality sign is \(\geq\), we shade above the solid boundary line. This shaded region includes all points that make the inequality true. In contrast, if the inequality were \(y < 3x + 9\), we would shade below a dashed boundary line. It's important to always shade in the direction that coincides with the inequality sign, which shows the area representing all potential solutions.