Problem 17 - Algebra and Trigonometry

Question

Sketch the region given by the set. \(\\{(x, y)| | x |>4\\}\)

Step-by-Step Solution

Answer

Shade the region to the right of \(x=4\) and to the left of \(x=-4\).

Step 1: Understand the Inequality

The exercise asks us to sketch the region defined by the inequality \(|x| > 4\). This inequality means that the absolute value of the x-coordinate must be greater than 4. This implies two conditions: either \(x > 4\) or \(x < -4\).

Step 2: Set Up the Boundaries

The boundaries for these conditions are the vertical lines \(x = 4\) and \(x = -4\). These lines serve as the thresholds where the condition changes from false to true.

Step 3: Shade the Significant Regions

Since \(|x| > 4\) means x is either greater than 4 or less than -4, we shade the regions to the right of the line \(x = 4\) and to the left of the line \(x = -4\).

Step 4: Draw the Graph

Draw two vertical lines, one at \(x = 4\) and another at \(x = -4\), and shade the areas to the right of \(x = 4\) and to the left of \(x = -4\). Make sure not to include the area between these lines.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions that have an absolute value, like \(|x| > 4\). The absolute value is a measure of distance from zero on a number line, meaning it’s always non-negative. When you see an inequality like this, it's talking about a range of values where the expression is true. For \(|x| > 4\), it means we're looking for values of \(x\) that are farther than 4 units away from zero. This results in two separate inequality conditions: one where \(x\) is greater than 4 and another where \(x\) is less than -4. So, any solution for \(|x| > 4\) must satisfy either \(x > 4\) or \(x < -4\). Understanding these conditions helps in visualizing the graph, as they indicate which part of the number line, and subsequently the graph, will be relevant.

Coordinate Plane

A coordinate plane is a two-dimensional surface on which points are defined by a pair of numbers (coordinates). Usually, we refer to it as the \(xy\)-plane, where horizontal axis (x-axis) represents all possible values of \(x\) and the vertical axis (y-axis) represents all possible values of \(y\). Each point on this plane is identified by an ordered pair \( (x, y)\). This plane allows us to visually explore relationships and distances between points. In our context, we will use it to plot regions that satisfy a specific inequality like \(|x| > 4\). Knowing how to navigate and interpret the coordinate plane is essential for graphing solutions to inequalities, as it helps in determining which areas should be shaded or plotted.

Graphing Regions

Graphing a region means identifying a set of points on the coordinate plane that satisfy a particular condition or inequality. For example, the solution of an inequality like \(|x| > 4\) creates two distinct regions. To determine these regions, we first look at the points that meet the condition, like those greater than 4 and less than -4 on the x-axis. Once the appropriate lines are drawn at the boundary points (in this case \(x=4\) and \(x=-4\)), the regions to the right of \(x=4\) and left of \(x=-4\) are shaded. This shading visually indicates where all valid solutions are located. By graphically representing the inequality, it becomes easier to understand which parts of the plane meet the criteria of \(|x| > 4\). It’s a powerful visual tool that aids in comprehending abstract mathematical relationships.

Vertical Lines

Vertical lines on the coordinate plane are special because they do not change in the y-direction. Instead, they represent all possible values of \(y\) at a specific \(x\). For our problem, the lines \(x = 4\) and \(x = -4\), are vertical lines. These lines act as boundaries, separating the regions of the inequality \(|x| > 4\). A vertical line is best represented by the equation \(x = c\), where \(c\) is a constant. In the context of our graph, each vertical line marks the point where the inequality stops being satisfied. Neither of these lines is included in our shaded regions because the inequality is strict (greater than, not greater than or equal to). So, when graphing inequalities, remember that a vertical line means we are not considering any change in \(y\), just focusing on the division of \(x\)-values.