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Linear inequalities are relations between two expressions that may not be equal but have a distinct order, defined by symbols like \textless, \textgreater, \textless=, or \textgreater=. Unlike a linear equation, which has a single line with points representing solutions, a linear inequality represents a set of points on one side of the line that fulfill the inequality condition.

A linear inequality in two variables, such as \(x - 3y \leq 6\), defines a half-plane region in a coordinate system in which every point in that region satisfies the inequality. The first crucial step in graphing a linear inequality is to form the equivalent equation of the boundary line by replacing the inequality sign with an equal sign. This allows us to understand where the 'divide' between solutions and non-solutions lies on the graph.

To sketch the graph of a line quickly, the slope-intercept form is incredibly useful. It's written as \(y = mx + b\), where \(m\) stands for the slope of the line and \(b\) indicates the y-intercept, which is where the line crosses the y-axis.

The slope, \(m\), tells us how steeply the line rises or falls as we move along it. A positive slope means the line goes upward from left to right, while a negative slope means it goes downward. The slope is determined by the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. For our inequality, we have a slope of \(\frac{1}{3}\), indicating that for every step we move to the right along the x-axis, we move up \(\frac{1}{3}\) units along the y-axis. The y-intercept, \(b\), in our inequality is -2, showing that our line crosses the y-axis at \((0, -2)\).

The shading on a graph of a linear inequality indicates the set of solutions to the inequality. It showcases the region of the graph where any point you pick will satisfy the inequality statement. After graphing the boundary line, we use a test point to determine which side of the line to shade.

Essentially, if the inequality symbol includes equal to, \(\leq\) or \(\geq\), the boundary line is solid because points on the line are included in the solution set. If the inequality does not include equality, such as \(\textless\) or \(\textgreater\), the boundary line is dashed, excluding the points on the line. In our exercise, because we have a 'less than or equal to' sign, \(\leq\), we represent the line with a dashed style and shade below it, signifying that all the points in this region, including those on the line, make the inequality true.