Problem 30 - Intermediate Algebra

Question

Graph \(|x-y|>2\).

Step-by-Step Solution

Answer

The graph consists of two shaded half-planes outside the region between \(x = y+2\) and \(x = y-2\), with both lines dashed.

Step 1: Understand the Inequality

The inequality \(|x-y| > 2\) means the absolute difference between \(x\) and \(y\) exceeds 2. This indicates that the distance between \(x\) and \(y\) on the number line is greater than 2.

Step 2: Break Down the Absolute Value

The inequality \(|x-y| > 2\) can be broken down into two cases: (1) \(x - y > 2\) and (2) \(x - y < -2\). These inequalities divide the graph into regions where these conditions are true.

Step 3: Solve the First Case

For the first case, \(x - y > 2\), rearrange it to \(x > y + 2\). This defines a half-plane in the coordinate system above the line \(x = y + 2\).

Step 4: Solve the Second Case

For the second case, \(x - y < -2\), rearrange it to \(x < y - 2\). This defines a half-plane in the coordinate system below the line \(x = y - 2\).

Step 5: Graph the Solution

To graph this system, draw the lines \(x = y + 2\) and \(x = y - 2\) in the coordinate plane. Since it is a strict inequality (\(>\)), these lines should be dashed. Shade the regions above \(x = y + 2\) and below \(x = y - 2\) where the inequalities hold.

Key Concepts

Absolute Value

The concept of absolute value is crucial when dealing with inequalities featuring expressions like \(|x-y| > 2\). \(|x-y|\) represents the absolute difference between two numbers, \(x\) and \(y\). In simpler terms, it's the distance between these two numbers on a number line, without considering which is larger. Absolute value transforms any negative result of subtraction into a positive one, reflecting the notion of 'distance'.
For instance, whether \(x-y\) equals 3 or -3, the absolute value result is 3.
Understanding this helps simplify inequalities by breaking them into manageable cases, such as \(x-y > 2\) and \(x-y < -2\). This breakdown is essential to investigate the solutions visually on a graph.

Coordinate Plane

A coordinate plane is a two-dimensional plane formed by the intersection of two perpendicular number lines: the x-axis and the y-axis. This plane is fundamental for graphing equations and inequalities, as it provides a visual framework where relationships between numbers can be easily depicted.
Each point on the plane is represented by a pair of numbers \((x, y)\), showing its exact location relative to the axes. In the problem, as we graph the inequalities of \(x > y + 2\) and \(x < y - 2\), the lines \(x = y + 2\) and \(x = y - 2\) divide the coordinate plane into distinct regions.

• Dash the lines \(x = y + 2\) and \(x = y - 2\) to indicate that points on the lines are not included in the solution, due to the \(>\) and \(<\) inequalities.
• Shading above and below these lines corresponds to the solutions for the inequalities.

Using the coordinate plane helps one visualize where the solutions to \(|x-y| > 2\) lie, giving a clear visual representation of which regions satisfy the inequality.

Graphing Solutions

Graphing solutions involves illustrating the possible values of \(x\) and \(y\) that satisfy a given equation or inequality on a coordinate plane. For the inequality \(|x-y| > 2\), it encompasses multiple steps, beginning with understanding how to represent and manipulate the absolute value expression.
Once reduced to the simpler inequalities \(x > y + 2\) and \(x < y - 2\), graph these equations on the plane. Dash the lines because the strict inequality excludes exact points on them.

• These lines, \(x = y + 2\) and \(x = y - 2\), serve as boundaries between possible solution areas and non-solution areas.
• Next, consider the region that satisfies each inequality: shade the area above the line \(x = y + 2\) and below the line \(x = y - 2\). This approach does not just conclude the math description; it clearly maps out the behaviors and relationships enforced by the inequality.

Graphing with care helps in thoroughly understanding which segments of \(x\) and \(y\) fulfill the inequality \(|x-y| > 2\), making it not just a calculation but a visual assertion of the solution.