Problem 20 - Algebra and Trigonometry

Question

Sketch the region given by the set. \(\\{(x, y)| | x |>2 \text { and }|y|>3\\}\)

Step-by-Step Solution

Answer

The region consists of four disjoint parts outside the lines \(x = \pm 2\) and \(y = \pm 3\).

Step 1: Understanding the Set Notation

The set notation given is \(\{(x, y)| |x|>2 \text{ and } |y|>3\}\) which describes a region in the xy-plane. The condition \(|x| > 2\) means that x is either less than -2 or greater than 2. Similarly, the condition \(|y| > 3\) means that y is either less than -3 or greater than 3.

Step 2: Analyzing the x-Condition

For \(|x| > 2\), we split this into two parts: \(x < -2\) and \(x > 2\). These two inequalities create regions outside the vertical lines \(x = -2\) and \(x = 2\) respectively.

Step 3: Analyzing the y-Condition

For \(|y| > 3\), we similarly split this into two parts: \(y < -3\) and \(y > 3\). These two inequalities create regions outside the horizontal lines \(y = -3\) and \(y = 3\) respectively.

Step 4: Combining Conditions to Find the Final Region

The specified region is where both conditions are satisfied simultaneously. Thus, it is the combination of regions outside both the lines \(x = -2\) and \(x = 2\), as well as \(y = -3\) and \(y = 3\). This results in four disjoint rectangular regions in the xy-plane.

Step 5: Sketching the Region

To sketch these regions, draw vertical dashed lines at \(x = -2\) and \(x = 2\) and horizontal dashed lines at \(y = -3\) and \(y = 3\). Shade the four resulting regions outside these lines: above and below the horizontal lines, to the left and right of the vertical lines.

Key Concepts

Inequalities in Algebra

Inequalities in algebra are expressions that show the relation between two values. Instead of expressions being equal as in equations, inequalities demonstrate if one value is larger or smaller than another. In our exercise, we have absolute value inequalities which include expressions like \(|x| > 2\) and \(|y| > 3\). These express that the distance of x and y from 0 is greater than 2 and 3, respectively.

To handle absolute value inequalities, we split them into two separate inequalities: \(x > 2\) or \(x < -2\) for \(|x| > 2\), and similarly, \(y > 3\) or \(y < -3\) for \(|y| > 3\). This way, we create solutions for x and y that fulfill the condition of being outside a given range.

• An inequality like \(a < b\) implies everything less than b.
• An inequality as \(a > b\) signifies everything greater than b.
• Combining inequalities with "and" means both conditions must be true at the same time in a region.

Leonor way to visualize these inequalities is by plotting them, turning these algebraic expressions into visual forms.

xy-plane Regions

The xy-plane, also called the Cartesian plane, is where we graphically represent the solutions of equations or inequalities involving two variables, x and y. Our task involves identifying specific regions bounded by inequalities.

The given conditions \(x < -2\) and \(x > 2\) mean that the x-values are in regions outside the vertical lines at these points, defined by \(|x| > 2\). Similarly, \(y < -3\) and \(y > 3\) define y-regions outside the horizontal bounds given by \(|y| > 3\).

• The xy-plane is vast, containing infinitely many points that denote all possible values of x and y.
• Regions in the plane can be visualized as areas confined by the boundaries.
• The combination of these designated regions gives us the solutions of inequalities in an easy-to-grasp visual format.

These insights make it easier to identify which areas satisfy the given conditions, simplifying complex inequalities into accessible graphics.

Sketching Graphs

Sketching graphs is a powerful tool to visualize mathematical concepts and inequalities. In our example, we're sketching to discover the region of the xy-plane that meets the conditions \(|x|>2\) and \(|y|>3\). Here's a step-by-step to make this clearer:

• Start by plotting the critical lines: draw vertical dashed lines at \(x = -2\) and \(x = 2\), and horizontal dashed lines at \(y = -3\) and \(y = 3\).
• Highlight or shade the regions that satisfy \(x < -2\) or \(x > 2\), and \(y < -3\) or \(y > 3\).
• The combination of areas outside these lines represents the region described by the set.

Breaking these tasks into simple parts allows us to tackle complex problems. It provides a complete picture that shows all possible solutions, guiding us through mathematical exploration with a methodical approach.