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Inequalities in two variables are expressions that relate two variables in an inequality format, such as \( y < x + 2 \) or \( x \geq 3 \). In this type of inequality, the solution is not a single point, but rather a collection of points that satisfy the condition described by the inequality.
The set \( \{ (x, y) | x \leq 0 \} \) is an example of an inequality where we focus on the relationship between the variables **x** and **y**. The inequality specifically tells us about the **x** values, stating that the **x** coordinate should be less than or equal to zero.
Because these inequalities cover regions or areas, we often use graphing to visually represent which points, or tuples, satisfy the inequality condition on a coordinate plane. This helps in visually understanding and interpreting these regions, especially when dealing with multiple inequalities simultaneously. Inequalities like \( x \leq 0 \) define regions to one side of a line or curve that serves as the boundary of the solution set.

Graphing inequalities involves plotting regions on a coordinate plane that satisfy the given inequality condition. Unlike equations that result in a specific line or curve, inequalities result in shaded regions that represent all solutions to the inequality.
When graphing an inequality, like \( x \leq 0 \), you begin by considering the related equation \( x = 0 \), which indicates a vertical line. To graph \( x \leq 0 \), you first draw this boundary line. Whether this line is solid or dashed depends on the inequality:

• A **solid line** is used for inequalities that include equal to (\( \leq \) or \( \geq \)).
• A **dashed line** indicates strict inequalities (\( < \) or \( > \)), showing that the line itself is not part of the region.

After drawing the line, you determine which side of the line contains the solutions. For \( x \leq 0 \), this is the half-plane to the left of the line, including all points with **x** values less than or equal to zero. This region is shaded or marked, indicating all possible solutions to the inequality.

Vertical line sketching is a straightforward process used in graphing inequalities and equations involving only **x**. When asked to sketch the vertical line for \( x = 0 \), you are essentially drawing the y-axis.
To draw this line, locate the origin \((0,0)\) on the coordinate plane and draw a straight line that extends upwards and downwards, perfectly vertical, going through the origin. This line is key when graphing the inequality \( x \leq 0 \), as it forms the boundary. Because the inequality is \( x \leq 0 \), the line must be solid, showing that points on the line \( x = 0 \) are included in the solution set.
Understanding how to graph vertical lines is crucial, especially in problems involving rectangles or boundaries where one or more sides are defined by such lines. The simplicity of a vertical line sketch should not be underestimated, as it forms the backbone of evaluating and depicting various regions defined by inequalities.