Problem 60 - Calculus with Concepts in Calculus

Question

Sketch the region in the plane satisfying the given conditions. \(x \leq 3\) and \(y \leq 2\)

Step-by-Step Solution

Answer

The region is a rectangle bounded by \( x = 3 \) and \( y = 2 \), extending to the left and downward.

Step 1: Understand the Inequalities

We are given two inequalities, namely, \( x \leq 3 \) and \( y \leq 2 \). These inequalities describe half-planes on a Cartesian coordinate system. The inequality \( x \leq 3 \) represents all the points to the left of and including the vertical line \( x = 3 \). Similarly, the inequality \( y \leq 2 \) represents all the points below and including the horizontal line \( y = 2 \).

Step 2: Draw the Boundary Lines

On a Cartesian coordinate plane, draw the vertical line \( x = 3 \). This line is solid because the inequality includes the boundary (\( x \leq 3 \)). Next, draw the horizontal line \( y = 2 \), also as a solid line because the inequality is \( y \leq 2 \). Both lines help define the boundaries of the region.

Step 3: Identify and Shade the Region

The solution region that satisfies both inequalities is where their solutions overlap. Shade the half-plane to the left of the \( x = 3 \) line and below the \( y = 2 \) line. The overlap of these shaded regions is a rectangle extending infinitely left and down, bounded by the lines \( x = 3 \) and \( y = 2 \).

Step 4: Final Check and Label

Ensure the shaded region intersects only where both inequalities are satisfied: left of or on \( x = 3 \) and below or on \( y = 2 \). Remember to label the axes and shade lightly to clearly show the region of interest.

Key Concepts

Half-planes

When we talk about half-planes, we refer to the two parts into which a plane is divided by a straight line. In this exercise, each inequality shapes a specific half-plane. Let’s dissect this concept a bit more:

• The inequality \( x \leq 3 \) suggests a half-plane that includes every point to the left of the line \( x = 3 \). This encompasses the area up to, and including, the line itself.
• Similarly, \( y \leq 2 \) defines a half-plane that covers all points beneath the line \( y = 2 \), with the line included in this territory too.

Understanding how these half-planes operate gives clarity to the shaded area sought in the exercise, as it lies beside or below these boundary lines. Both half-planes won't extend beyond their respective lines in the directions specified.

Cartesian coordinate system

The Cartesian coordinate system serves as the stage where we sketch and analyze the inequalities. It's a grid formed by horizontal and vertical axes, traditionally labeled \( x \) and \( y \). This grid enables the precise plotting of equations and inequalities.

• The vertical axis is the \( y \)-axis, and the horizontal one is the \( x \)-axis. Each point in this system has a pair of values indicating positions along these axes \((x, y)\).
• This system provides the groundwork to visualize data, equations, and inequalities, such as \( x \leq 3 \) and \( y \leq 2 \), making it an invaluable tool in understanding geometric relationships.

For our exercise, knowing where to draw each line and shade each region on the Cartesian plane aids significantly in interpreting the inequalities delineated. It offers a clear and structured layout to show the points involved.

Boundary lines

Boundary lines are critical to constructing solutions in inequalities. They define the limits of half-planes on the Cartesian plane.

• In our case, the boundary line for \( x \leq 3 \) is the vertical line on the graph where \( x = 3 \). As the inequality includes "equal to," it’s represented by a solid line.
• Similarly, the boundary line for \( y \leq 2 \) is horizontal at \( y = 2 \). Like the \( x \)-boundary, it’s also solid due to the "equal to" component.
• These lines intersect creating a boundary for where the solutions to the inequalities overlap.

Where these boundary lines intersect in the Cartesian plane outlines a specific region, aiding in the visualization and comprehension of where both inequalities hold true in a shared graphical space.