Problem 23 - Precalculus Mathematics for Calculus

Question

Sketch the region given by the set. $$\\{(x, y) | x \geq 3\\}$$

Step-by-Step Solution

Answer

The region is to the right of the vertical line \( x = 3 \).

Step 1: Interpret the Inequality

The set \( \{(x, y) | x \geq 3\} \) represents all points \( (x, y) \) in the Cartesian plane where the \( x \)-coordinate is at least 3. This means for any \( x \)-value, it must be greater than or equal to 3, while the \( y \)-value can be any real number.

Step 2: Identify the Boundary Line

The boundary of this region is the vertical line \( x = 3 \). This line represents the smallest \( x \)-value of the region, where \( x \) is equal to 3.

Step 3: Determine Which Side of the Line Represents the Inequality

Since the inequality is \( x \geq 3 \), the region of interest is to the right of the line \( x = 3 \). Any \( x \)-value greater than or equal to 3 is included in the region.

Step 4: Sketch the Region

First, draw the vertical line \( x = 3 \) on the Cartesian plane. Mark this line by either a solid line to indicate that it is part of the solution (due to the 'equal' part of \( \geq \)). Then, shade the entire region to the right of this line. This represents all \( x \)-values greater than or equal to 3, for any \( y \)-value.

Key Concepts

Cartesian plane

The Cartesian plane is a two-dimensional surface where we can plot points, lines, and regions using two perpendicular number lines: the x-axis and the y-axis. This system helps us visualize mathematical relationships and solve problems like inequalities.

On this plane, points are specified using coordinates, written as \(x, y\). The x-coordinate indicates a position along the horizontal axis, while the y-coordinate indicates a position along the vertical axis.

Using this system, mathematicians and students can visually interpret data and relationships between variables. When dealing with inequalities like \(x \geq 3\), the focus is on identifying a region on this plane that meets specific criteria.

Boundary line

Every inequality has a boundary line that helps divide the plane into different regions. In the exercise \(x \geq 3\), the boundary line is the vertical line \(x = 3\).

This line acts as a threshold that classifies which parts of the plane satisfy the inequality.

• For the inequality \(x \geq 3\), the boundary line \(x = 3\) itself is included in the region. Hence, it is drawn as a **solid** line.
• If the inequality were strict, like \(x > 3\), the boundary would be represented by a **dashed** line, indicating that points on the line are not included in the solution set.

The use of a boundary line simplifies determining which side of the line contains the solution by providing an easy visual cue.

Inequality sketching

Inequality sketching involves visually representing the solution set of an inequality on the Cartesian plane. This technique helps in understanding which regions satisfy the given conditions.

For the inequality \(x \geq 3\):

• Start by drawing the boundary line \(x = 3\) with a solid line to show that \(x = 3\) is part of the solution.
• Then, shade the region to the right of the line. This area represents all points where the x-coordinates are greater than or equal to 3, while y can be any real number.

These sketches are powerful tools for visual learners and provide a clear depiction of how inequalities divide the plane into regions, assisting in deeper comprehension of algebraic expressions.