Problem 58 - Precalculus with Limits

Question

In Exercises 53-60, write a system of inequalities to describe the region. Parallelogram: vertices at (0,0),(4,0),(1,4),(5,4)

Step-by-Step Solution

Answer

The system of inequalities that describe the region of the parallelogram are: \(0 \leq y \leq 4\), \(y \geq 4x\) and \(y \leq 4x -16\)

Step 1: Determine the equations of the lines

The vertices of the parallelogram are (0,0),(4,0),(1,4), and (5,4). Any two points can determine a line. The slope of the line can be determined by \((y_2 - y_1) / (x_2 - x_1)\). The equation of the line going through points (0,0) and (4,0) is \(y = 0\), The equation of the line going through points (0,0) and (1,4) can be found as follows:The slope (m) is \((4 - 0) / (1 - 0) = 4\).So, the line's equation is \(y = 4x\),The equation of the line going through points (1,4) and (5,4) is \(y = 4\), The equation of the line going through points (4,0) and (5,4) is as follows:The slope (m) is \((4 - 0) / (5 - 4) = 4\).So, the line's equation is \(y = 4x - 16\).

Step 2: Formulate the inequalities

Now that the equations of the four sides of the parallelogram are determined, the inequalities that represent the region inside and on the sides of the parallelogram can be written. The region is above the x-axis and below the line \(y = 4\), so \(0 \leq y \leq 4\). It is to the right of the line \(y = 4x\) and to the left of the line \(y = 4x - 16\), so \(y \geq 4x\) and \(y \leq 4x -16\).

Key Concepts

Understanding the Slope of a Line

In geometry, the slope of a line is a fundamental property that helps identify the direction and inclination of the line on a coordinate plane. It's crucial for differentiating lines and ensuring exact positioning of various shapes, like parallelograms. To find the slope, use the formula: \( (y_2 - y_1) / (x_2 - x_1) \), which represents the change in the y-values divided by the change in the x-values between two distinct points.

• The slope describes whether a line ascends, descends, or remains flat as it moves from left to right on a graph.
• A positive slope indicates an upward trend, while a negative slope reflects a downward trend. A zero slope represents a horizontal line.

For example, in the given parallelogram, the lines between points such as (0,0) to (1,4) and (4,0) to (5,4) have the same slope of 4, indicating they are parallel and rise steeply along the y-axis as x increases. Understanding slopes aids in forming accurate systems of equations and inequalities for the task.

Characteristics of a Parallelogram

A parallelogram is a special type of quadrilateral with opposite sides that are parallel and equal in length. It has unique properties that help in identifying and plotting graphs for various mathematical problems.

• In a parallelogram, opposite sides are equal. Hence, in the given problem, sides such as (0,0) to (4,0) and (1,4) to (5,4) are parallel and also have equal length.
• Opposite angles in a parallelogram are equal, which can assist in verifying if the constructed lines form a parallelogram as required.

By plotting the vertices given: (0,0), (4,0), (1,4), and (5,4), and connecting them with straight lines, we confirm the property of parallel lines (check if the slope of the lines between opposite points is equal). This geometric property strengthens understanding when setting up systems of inequalities, as it tells us about relationships between lines and regions they enclose.

Introduction to Linear Equations

Linear equations are statements of equality involving linear expressions. They typically involve variables raised to the first power and do not include exponential, trigonometric functions, or variables multiplied together. Linear equations are pivotal in describing relationships between two variables.The general form of a linear equation is \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point at which the line crosses the y-axis.

In the exercise, several linear equations like \( y = 0 \) or \( y = 4x - 16 \) have been derived. Each equation describes one boundary of the parallelogram. Subsequent use of inequalities, such as \( 0 \leq y \leq 4 \), further refines these equations to capture the region inside the parallelogram.Linear equations and their graphical representation are fundamental tools for understanding and solving systems of inequalities. They establish a foundation for visualizing mathematical relationships on coordinate planes. By converting real-world scenarios like shapes and boundaries into linear equations, students can explore and solve complex mathematical concepts effectively.