Problem 40 - Algebra 2

Question

CHALLENGE Graph \(|y| < x\)

Step-by-Step Solution

Answer

Shade the region between the lines \\(y=x\\) and \\(y=-x\\); exclude the lines.

Step 1: Understand the Equation Components

Identify the components of the inequality \(|y| < x\). The expression \(|y|\) indicates the absolute value of \(y\), and \(x\) is a linear expression representing a horizontal line.

Step 2: Identify Absolute Value Graph Features

The inequality \(|y| < x\) signifies that the distance of \(y\) from zero must be less than \(x\). In other words, \(-x < y < x\). This represents the band or region between two lines: \(-x\) and \(x\).

Step 3: Determine Boundary Lines

Graph the boundary lines \(y = x\) and \(y = -x\). These lines form the outer bounds of the region where \(|y| < x\).

Step 4: Shade the Solution Region

The inequality \(|y| < x\) involves shading the region between \(y = x\) and \(y = -x\), not including the lines themselves because the inequality is strict (<).

Step 5: Verify Graph Behavior

The shaded region demonstrates points where \(|y| < x\). Therefore, every point in the shaded area between these lines is a solution to the inequality.

Key Concepts

Graphing Inequalities

When graphing inequalities, the first thing you need to understand is that you're not just plotting a line but rather a region of values that satisfy the inequality. Unlike single solutions in equations, inequalities have a range of solutions. This range is often represented as a shaded area on the graph. The ideal outcome is to visually represent where any solutions can exist.

• Draw the boundary line that is formed by turning the inequality into an equation, such as replacing "<" with "=".
• Decide whether the boundary line should be solid or dashed. A solid line is used for ≤ or ≥ (indicating inclusion), and a dashed line is for < or > (indicating exclusion).
• Shade the region that satisfies the inequality — this is your solution set.

Graphing inequalities helps to see the multiple solutions visually and can often make understanding complex constraints simpler.

Linear Equations

Linear equations play a foundational role in graphing inequalities. A linear equation is an equation between two variables that results in a straight line when graphed. This concept is key in understanding inequalities like \(|y| < x\). For example, both lines defined by the equation \(y = x\) or \(y = -x\) are straight lines.

Key properties of linear equations include:

• The slope, which determines the angle and direction of the line.
• The y-intercept, which indicates where the line crosses the y-axis.

In the context of the inequality from the exercise, knowing how to graph the lines \(y = x\) and \(y = -x\) accurately ensures the correct identification of the boundary of the solution region. Linear equations help you structure inequalities systematically for easy interpretation and graphing.

Solution Region

The solution region in graphing an inequality refers to the entire area on a graph where the inequality holds true. For the inequality \(|y| < x\), the solution region is the area between the lines \(y = x\) and \(y = -x\). Graphically, this means shading the entire strip between the two boundary lines without including the lines themselves since the inequality is a strict one ("<" rather than "≤").

Important aspects to remember include:

• The boundary lines are not part of the solution — represented by dashed lines.
• Any point in the shaded area is a valid solution.
• The solution area visually represents all possible values that satisfy the inequality.

Understanding the solution region is crucial for verifying that the right areas are marked, ensuring the inequality is represented correctly.

Inequality Graphing Steps

Graphing inequalities typically involves a set of structured steps to ensure that the solution is accurately represented. Let's review these inequality graphing steps within the context of \(|y| < x\):

1. Turn the inequality into an equation to find the boundary lines.
2. Graph the boundary lines (|\y = x\} and \(y = -x\)). Use dashed lines because of the strict inequality.
3. Identify the region to shade. For \( |y| < x \), it's the area between \(y = x\) and \(y = -x\).
4. Verify by picking a test point in the shaded region to ensure it satisfies the inequality.

Following these steps in order helps to accurately plot the solution region and fully understand the relationship between the mathematical expression and its visual representation. This systematic approach ensures that errors are minimized, and the correct solution is highlighted effectively.