Problem 16 - Precalculus

Question

Graph the given relation. $$ \\{(x, y) \mid x \leq 3\\} $$

Step-by-Step Solution

Answer

Graph a solid vertical line at \( x = 3 \) and shade to the left.

Step 1: Understand the Relation

The relation \( \{(x, y) \mid x \leq 3\} \) involves all points \((x, y)\) on a plane where the x-coordinate is less than or equal to 3. This indicates that the graph will include every point to the left of the vertical line \( x = 3 \).

Step 2: Determine the Graph Boundary

The boundary for this graph is the vertical line \( x = 3 \). Since the relation includes \( x = 3 \), this line is part of the solution set and will be drawn as a solid line.

Step 3: Identify the Included Region

Since we are considering \( x \leq 3 \), the region we need to shade includes all points to the left of the line \( x = 3 \). This includes all possible y-values for each x-value that satisfies \( x \leq 3 \).

Step 4: Sketch the Graph

To graph the relation, draw a solid vertical line at \( x = 3 \) on the coordinate plane. Then, shade the entire region to the left of this line to represent every point where the x-coordinate is less than or equal to 3.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and various shapes. It consists of two number lines:

• The horizontal line, or x-axis.
• The vertical line, or y-axis.

These axes intersect at a point called the origin, which has coordinates (0, 0). Points on the plane are defined by ordered pairs \(x, y\), where \x\ is the position along the horizontal axis and \y\ is the position along the vertical axis. This system allows us to locate any point by moving a certain distance along the x and y directions. Understanding how to navigate the coordinate plane is essential for graphing relations, as it forms the foundational grid on which we place our points and draw lines.

Vertical Line

A vertical line is a straight line that moves up and down on the coordinate plane. This line is characterized by a constant x-coordinate. As part of graphing exercises, understanding the formation and implication of vertical lines is key.
For example, the line \(x = 3\) is vertical because for any point on this line, the x-coordinate remains 3, regardless of the y-value. In our graphing relation, \(x = 3\) acts as the boundary of the shaded region. Vertical lines are visually represented as lines parallel to the y-axis and are integral in defining constraints or limits in graphing scenarios.

Inequalities in Graphing

When graphing inequalities like \(x \leq 3\), we are not just plotting a line; we are exploring a range of solutions that satisfy certain conditions. Inequalities expand on the idea of equations by including less than, greater than, or equal to comparisons. For the inequality \(x \leq 3\), it means every point with an x-value less than or equal to 3 is valid.
This involves using different line styles:

• A solid line indicates that the boundary itself is included, as with \(x = 3\) is in \(x \leq 3\).
• A dashed line is used when the boundary is not included (e.g., \(x < 3\)).

Inequalities in graphing help define regions, restrictions, and possible solutions much more vividly than a simple equation.

Shading Regions

Shading regions on a graph is a way to visually represent the solution set of an inequality. It illustrates all the points that satisfy the given conditions. When the relation is \(x \leq 3\), the shading will appear to the left of the vertical line \(x = 3\).
Follow these steps to determine the shaded region:

• First, draw the boundary line (solid for \(x \leq 3\)).
• Next, decide which side of the line satisfies the inequality.
• Then, proceed to shade that entire area.

Shading is a crucial tool in inequalities as it distinguishes which parts of the graph comply with the solution set, thus providing a clear visual interpretation of what the inequality represents on the coordinate plane.