Problem 32 - Algebra and Trigonometry

Question

\(19-32\) 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 disjointed areas in each quadrant outside the rectangle bounded by \(x = -2, x = 2, y = -3, y = 3\).

Step 1: Interpret the inequalities for x and y

The condition \(|x| > 2\) implies that either \(x > 2\) or \(x < -2\). This describes the region outside the vertical lines \(x = 2\) and \(x = -2\). Similarly, \(|y| > 3\) implies that either \(y > 3\) or \(y < -3\), describing the region outside the horizontal lines \(y = 3\) and \(y = -3\).

Step 2: Identify the regions described

For \(|x| > 2\), the region consists of the areas to the left of \(x = -2\) and to the right of \(x = 2\). For \(|y| > 3\), the region consists of the areas below \(y = -3\) and above \(y = 3\).

Step 3: Determine the intersection of regions

The resulting region is where both conditions \(|x| > 2\) and \(|y| > 3\) hold simultaneously. This results in four disjoint quadrants located in the plane: upper left (\(x < -2\), \(y > 3\)), upper right (\(x > 2\), \(y > 3\)), lower left (\(x < -2\), \(y < -3\)), and lower right (\(x > 2\), \(y < -3\)).

Step 4: Sketch the region

Draw the Cartesian plane and sketch the boundary lines:\(x = -2\), \(x = 2\), \(y = -3\), and \(y = 3\). Shade the areas external to these lines. The shaded regions do not lie inside the central rectangle defined by these lines. It includes: above \(y=3\) for \(x>2\) and \(x<-2\); below \(y=-3\) for \(x>2\) and \(x<-2\).

Key Concepts

Cartesian Plane

The Cartesian Plane is a two-dimensional plane where we can plot points, lines, and regions using coordinates. Each point on this plane is described by an ordered pair \(x, y\), where \x\ represents the horizontal distance and \y\ represents the vertical distance from the origin.

To help visualize equations and inequalities, the Cartesian Plane is split into four sections called quadrants, which we will explore further. It is named after René Descartes, a French mathematician and philosopher, who helped develop this concept. Understanding the Cartesian Plane is essential for graphing various mathematical expressions, especially inequalities.

Absolute Value

Absolute Value refers to the distance of a number from zero on the number line, regardless of direction. It is always non-negative.

For any real number \x\, the absolute value is denoted by \|x|\. If \x\ is positive or zero, the absolute value is \x\ itself; if \x\ is negative, the absolute value is \x\ made positive.

• For example, \|3|\ = 3\.
• Similarly, \|-3|\ = 3\.

In the context of inequalities, such as \( |x| > 2 \), this indicates two scenarios: either \( x > 2 \) or \( x < -2 \). This concept is crucial as it helps define boundaries and regions on the Cartesian Plane.

Graphing Inequalities

Graphing Inequalities involves shading a region on the Cartesian Plane to show where the inequality holds true. Inequalities are different from equalities because there are more solutions than a single line or point.

To effectively graph an inequality, you need to:

• Identify the boundary line(s) that form the limits of the inequality.
• Decide whether the lines themselves are part of the solution set. This usually depends on whether the inequality sign is strict (\< or \(\)>) or inclusive (\leq or \(\geq\)).
• Shade the region where the conditions of the inequality are satisfied.

For example, with inequalities like \( |x| > 2 \) and \(|y| > 3\), you are graphing the outside of two sets of parallel lines on the Cartesian Plane.

Quadrants in Coordinate Plane

In the Cartesian Plane, each point falls into one of four quadrants or sits on one of the axes. Each quadrant is defined by the signs of the \(x\) and \(y\) coordinates:

• Quadrant I: Both \(x\) and \(y\) are positive.
• Quadrant II: \(x\) is negative, \(y\) is positive.
• Quadrant III: Both \(x\) and \(y\) are negative.
• Quadrant IV: \(x\) is positive, \(y\) is negative.

When graphing the solution to inequalities like \( |x| > 2 \) and \(|y| > 3\), certain quadrants are more relevant. We look for areas outside the central rectangle created by the lines \(x = 2\), \(x = -2,\), \(y = 3,\), and \(y = -3\). This results in regions in Quadrants II, IV, I, and III beyond these lines, forming disconnected sections of our interest.