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When we talk about linear inequalities, we're referring to expressions that show a relationship of inequality—like greater than or less than—between two algebraic expressions that are linear. These expressions usually involve a variable, such as 'x' or 'y', which can take on various values.

In our exercise, we come across the linear inequality, \(y > \frac{1}{3} x\). Correctly interpreting this inequality is the first crucial step before graphing. Unlike an equation, an inequality does not express an exact relationship where 'y' equals one third of 'x', but rather a range of possibilities where 'y' is greater than one third of 'x'.

Linear inequalities are used to represent a wide range of real-life situations where we work with ranges or intervals, not just fixed numbers. Understanding how to graph these inequalities is essential for visualizing the range of possible solutions.

The coordinate system, often referred to as the Cartesian plane, is a two-dimensional plane where each point is uniquely determined by a pair of numerical coordinates. These coordinates are typically written as ordered pairs (x, y). The horizontal axis is known as the x-axis, and the vertical axis is the y-axis.

Graphing linear inequalities requires a firm understanding of the coordinate system since we use it to portray the relationship described by the inequality. As seen in the exercise, the line created by the equation \(y = \frac{1}{3}x\) can be graphed on this coordinate system. Each point on the line is a possible solution to the equivalent equation. In the context of the inequality, this line acts as a boundary to help visualize which part of the plane satisfies the condition posed by the inequality.

Inequality notation provides a way to express the relative value of two expressions. Common inequality symbols include \(<\) (less than), \(>\) (greater than), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to).

In our exercise, the symbol \(>\) signifies 'greater than', indicating that the 'y' values we're looking for are above the value of \(\frac{1}{3}x\) for any given 'x'. This directional information is critical—it tells us not only which side of the line to shade on the graph but also helps in understanding the range of possible values that 'y' can have related to 'x'.

When graphing, we utilize different lines to denote strict inequalities versus non-strict inequalities. A dashed line, as used in our exercise, indicates that points on the line itself are not included in the solution set, while a solid line means that those points are included.