Problem 38 - Calculus Single Variable

Question

Sketch the given region. \(\\{(x, y):|x| \leq 5,|y|>2\\}\)

Step-by-Step Solution

Answer

A vertical strip between x = -5 and x = 5, excluding the band between y = -2 and y = 2.

Step 1: Identify the Constraints for x

The given region specifies the constraint \(|x| \leq 5\). This means that the x-coordinates must be within -5 and 5, inclusive. Graphically, this corresponds to a vertical strip or band that stretches horizontally from x = -5 to x = 5.

Step 2: Identify the Constraints for y

The constraint \(|y| > 2\) tells us which y-coordinates are permissible. This means that y is not between -2 and 2, requiring two distinct horizontal regions: one where y > 2 and another where y < -2. These regions do not include y-values between -2 and 2.

Step 3: Combine the Constraints

Now, combine both constraints. Each point (x, y) must satisfy both \(|x| \leq 5\) and \(|y| > 2\). Therefore, our region will be within the vertical lines at x = -5 and x = 5, but will exclude the horizontal strip between y = -2 and y = 2.

Step 4: Sketch the Region

Begin by drawing the vertical lines x = -5 and x = 5 to define the boundary of x-values. Then, draw the horizontal lines y = -2 and y = 2 to establish the bounds of y-values that are excluded. Shade the areas above y = 2 and below y = -2 within the vertical strip, as these are the locations where all conditions are met.

Key Concepts

Inequalities

Inequalities are essential in mathematics, describing a range of possible values for a variable. They express conditions where one quantity is larger or smaller than another. In our exercise, we dealt with absolute value inequalities, specifically \(|x| \leq 5\) and \(|y| > 2\). Absolute value inequalities provide information on the distance of a number from zero on a number line.

• \(|x| \leq 5\) means that x can be any number from -5 to 5, inclusive. It forms a band of x-values.
• \(|y| > 2\) indicates that y must be greater than 2 or less than -2, forming two regions on a number line.

Understanding these inequalities is crucial because they allow us to visually and algebraically constrain the domain and range of a function or region. By sketching the solution, you can better grasp how these regions combine to specify the set of permissible points in the coordinate plane.

Graphing

Graphing inequalities allows us to visualize solutions, making complex relationships clearer. When graphing the region \({(x, y):|x| \leq 5,|y|>2}\), we highlight specific areas in the coordinate plane.

• The vertical strip from x = -5 to x = 5 demonstrates the boundary created by \(|x| \leq 5\). It's like a vertical band that's shaded.
• For \(|y| > 2\), we draw horizontal lines at y = 2 and y = -2. Regions outside these lines, both above y = 2 and below y = -2, are shaded.
• To visualize the solution, intersection areas fulfilling both conditions are shaded, excluding the unshaded horizontal strip within y = -2 to y = 2.

Different shading patterns help illustrate regions where inequalities overlap, supporting better understanding as you associate algebraic constraints with their graphical interpretations.

Coordinate System

Understanding the coordinate system is vital for effectively graphing inequalities. The Cartesian coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Every point in this system is defined by an ordered pair \(x, y\).

• The x-axis locates horizontal positions where x-values determine how far left or right a point is.
• The y-axis locates vertical positions, showing how high or low a point is based on y-values.
• Coordinates, given as \(x, y\), reveal the exact position on the plane, allowing complex constructs like inequalities to be visualized.

In our scenario, the coordinate system helps delineate regions meeting the inequality conditions. With x-values ranging from -5 to 5 and y-values excluding the -2 to 2 interval, the coordinate system visually conveys these relationships as a structured geometric space. This enhances comprehension by mapping algebraic solutions to real-world graphical representations.