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The coordinate plane is essential for graphing inequalities. It is a two-dimensional plane where each point is defined by a pair of numerical coordinates. These coordinates are written as \((x, y)\), where x is the horizontal position, and y is the vertical position. The plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). The origin is the point (0,0) where these axes intersect. Understanding the coordinate plane helps in placing points accurately and helps visualize how inequalities like \((x \leq \ 3)\) work in a spatial context. To graph an inequality, plot the variables accordingly along the x-axis and y-axis. This forms the basis for drawing boundary lines and shading regions.

For the inequality \((x \leq \ 3)\), the boundary line plays a vital role. The boundary line is drawn at the point where the inequality equals a specific value. In this case, it's a vertical line where x is always 3. We graph this line vertically at x = 3 on the coordinate plane. When dealing with inequalities, the type of boundary line matters. For \(<\) or \((\text{ > })\), we use a dashed line to show that points on the line are not included. For \((\text{ \leq })\) or \((\text{ \geq })\), we use a solid line to show that points on this line are included in the solution. Here, since our inequality is \((\text{ \leq })\), we draw a solid line at x = 3, indicating that the points on this line are part of the solution.

Shading the region is the final step in graphing inequalities on the coordinate plane. This step shows all the possible solutions to the inequality. For \((x \leq \ 3)\), we need to shade all the regions where x-values are less than or equal to 3. First, look at the boundary line at x = 3. Since the inequality symbol includes 'less than or equal to', we shade to the left of this line. Make sure to include the boundary line in your shading since x can equal exactly 3. The shaded area visually represents all the x-values that satisfy \((x \leq \ 3)\), making it clear and easy to understand.