Problem 20 - Precalculus: Functions and Graphs

Question

Sketch the graph of the system of Inequalities. $$\left\\{\begin{array}{l}|x| \geq 4 \\\|y| \geq 3\end{array}\right.$$

Step-by-Step Solution

Answer

The desired graph has shaded regions outside the vertical lines \(x = -4\) and \(x = 4\), and outside the horizontal lines \(y = -3\) and \(y = 3\).

Step 1: Interpret the Inequalities

The inequalities given are \(|x| \geq 4\) and \(|y| \geq 3\). The inequality \(|x| \geq 4\) means that \(x\) is either less than or equal to -4, or greater than or equal to 4. The inequality \(|y| \geq 3\) means that \(y\) is either less than or equal to -3, or greater than or equal to 3.

Step 2: Plot Horizontal Lines

Plot the horizontal lines at \(y = 3\) and \(y = -3\) on the graph. These lines are boundaries for the inequality \(|y| \geq 3\). The region of interest for \(y\) is outside the strip formed between these two horizontal lines.

Step 3: Plot Vertical Lines

Plot the vertical lines at \(x = 4\) and \(x = -4\) on the graph. These lines are boundaries for the inequality \(|x| \geq 4\). The region of interest for \(x\) is outside the strip formed between these two vertical lines.

Step 4: Shade the Regions

Shade the area outside of the region between the two horizontal lines. Also, shade the area outside of the region between the two vertical lines. The solution to the system of inequalities is the area that is shaded in both directions. This includes the four corner regions of the plane: top left, top right, bottom left, and bottom right.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities are a special type of inequality that involves the absolute value function. Absolute value, denoted by \(|x|\), represents the distance of a number from zero on the number line. Thus, when we say \(|x| \geq 4\), we mean that the distance of \(x\) from zero is at least 4. This leads to two scenarios:

• \(x \leq -4\)
• \(x \geq 4\)

These inequalities suggest that the acceptable values for \(x\) lie on either side of these points on a number line.
Similarly, for \(|y| \geq 3\), it implies:

• \(y \leq -3\)
• \(y \geq 3\)

The concept of absolute value inequalities can be visualized on a coordinate plane, where these constraints define specific regions where the solution lies beyond certain boundaries.

Coordinate Plane

The coordinate plane is a two-dimensional plane formed by the intersection of a vertical line, called the y-axis, and a horizontal line, called the x-axis. These two axes divide the plane into four quadrants. The coordinate plane is the basis for graphing equations and inequalities.When graphing the inequalities \(|x| \geq 4\) and \(|y| \geq 3\), the plane helps us draw horizontal and vertical lines that represent the boundaries of these inequalities. Each point in the plane can be represented by a pair of numbers \((x, y)\).

• The x-coordinate denotes how far to move horizontally from the origin (0,0).
• The y-coordinate indicates how far to move vertically.

Plotting inequalities involves shading specific regions of the plane where the solutions to the inequality exist. The coordinate plane allows us to visualize and identify these regions easily.

System of Inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. The solution to a system is the set of points that satisfy all inequalities in the system at the same time.In our exercise, the system involves \(|x| \geq 4\) and \(|y| \geq 3\). Each inequality describes a separate region on the coordinate plane:

• From \(|x| \geq 4\), we look for areas outside the vertical lines of \(x = -4\) and \(x = 4\).
• From \(|y| \geq 3\), we consider areas outside the horizontal lines of \(y = -3\) and \(y = 3\).

When considered together, the system's solution is the intersection of these regions—specifically, the four corner regions outside of both sets of boundary lines. These corners are where both conditions of the system are satisfied, giving us a clear graphical representation of the common solution area.