Problem 43 - Calculus Single Variable

Question

Sketch the given region. \(\\{(x, y):|x-y|<1, x \geq 4\\}\)

Step-by-Step Solution

Answer

The region is a vertical strip for \(x \geq 4\) between the lines \(y=x+1\) and \(y=x-1\).

Step 1: Understand the inequality

The given inequality is \(|x-y|<1\). This inequality can be rewritten as two separate inequalities: \(x-y<1\) and \(y-x<1\). These inequalities represent two lines in the xy-plane, specifically \(y = x+1\) and \(y = x-1\), and the region is the area between these two lines.

Step 2: Analyze x-value conditions

The condition \(x \geq 4\) confines our region of interest to lines and points only to the right of the line \(x=4\). Thus, we'll only consider the part of the region between \(y = x+1\) and \(y = x-1\) where \(x \geq 4\).

Step 3: Identify the region of interest

For \(x \geq 4\), the region of interest is the vertical strip between the lines \(x=4\) and \(x+1>y>x-1\). Points that satisfy the entire condition \(|x-y|<1\) are inside this strip, but also where the x-values start from 4 onward.

Step 4: Sketch the boundaries

Draw the lines \(y=x+1\) and \(y=x-1\) on a coordinate plane. Then draw the line \(x=4\). The area where these three regions intersect is located between the lines \(x=4\), \(y=x-1\), and \(y=x+1\).

Step 5: Shading the region

Shade the region to the right of \(x=4\) and in between the lines \(y=x-1\) and \(y=x+1\). This shaded part represents all the points \((x, y)\) that satisfy \(|x-y|<1\) and \(x \geq 4\).

Key Concepts

Coordinate Geometry

In the world of mathematics, coordinate geometry acts as a bridge between algebra and geometry by using a coordinate plane to represent equations and shapes. The xy-plane is a two-dimensional surface where you can plot points using pairs of numbers. Each point

• First Value: Corresponds to the x-coordinate (horizontal position)
• Second Value: Represents the y-coordinate (vertical position)

By plotting equations on this plane, we can visually represent different mathematical relationships.
Lines, shapes, and regions can all be defined using coordinate geometry. For instance, the equations of lines like

• \( y = x+1 \)
• \( y = x-1 \)

can be plotted on the xy-plane. These lines create different regions, and understanding these intersections is crucial for solving equations graphically.
In our example, the line \( x = 4 \) confines our region further, demonstrating how multiple equations work together in coordinate geometry to form specific boundaries and regions in the plane.

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a difference or calculation is compared to a number. The absolute value of a number is simply its distance from zero on the number line, regardless of direction, always being non-negative.
In the inequality

• \( |x-y| < 1 \)

we're essentially defining a range where the distance between x and y is less than 1. This means that y can only differ from x by less than 1, creating a "strip" region between two boundary lines.
When rewritten,

• \( x-y < 1 \)
• \( y-x < 1 \)

these inequalities convert into linear equations or boundary lines on the coordinate plane:

• \( y = x+1 \)
• \( y = x-1 \)

These define two parallel lines and the area in between them is the region satisfying the inequality. This is a good example of how absolute value inequalities translate into tangible areas on a graph.

Graphing Regions

Graphing regions involves sketching areas on the coordinate plane that satisfy a set of inequalities or conditions. To effectively find and shade these regions, we must first draw the boundary lines from the inequalities. These boundaries typically mark the edges of the regions.
For example, let's consider the inequalities

• \( y = x+1 \)
• \( y = x-1 \)

They form parallel lines on the plane. The inequality

• \( |x-y|<1 \)

describes the vertical strip between these two lines.
The additional condition

• \( x \geq 4 \)

limits this strip to the right of the line \( x = 4 \), meaning we only shade the region where the lines intersect, respecting all conditions. This shaded area visually represents all points \( (x, y) \) that satisfy both the absolute value inequalities and x-condition in the same plane. This process helps clarify and solve inequalities by "seeing" them on the graph.