Problem 59 - Precalculus with Limits

Question

In Exercises 53-60, write a system of inequalities to describe the region. Triangle: vertices at (0,0),(6,0),(1,5)

Step-by-Step Solution

Answer

The system of inequalities that represent the given triangular region is: \[y \geq 0\], \[y \leq 5x\] and \[y \leq -x + 6\].

Step 1: Identify the Lines

First, identify the lines that form the triangle. These are the lines connecting the given vertices: (0,0) to (6,0), (0,0) to (1,5), and (6,0) to (1,5). The equations of the lines are: \n Line 1: y = 0 (x-axis) \n Line 2: y = 5x \n Line 3: \(y = \frac{-5}{5}(x-6)\) which simplifies to y = -x + 6.

Step 2: Identify the Inequalities

Next, identify the inequalities that describe the given region. As the region is a triangle, the system of inequalities should represent the area within the triangle. The inequalities are therefore: \n \[y \leq 0\] (For the area below Line 1) \n \[y \leq 5x\] (For the area below Line 2) \n \[y \leq -x + 6\] (For the area below Line 3)

Step 3: Final Answer

Combine Steps 1 and 2 to write the system of inequalities that represent the given triangle region. The answer must state all three inequalities together.

Key Concepts

Linear Inequalities

Understanding linear inequalities is crucial for solving various mathematical problems, especially when dealing with relationships involving two variables that do not have equals signs. Consider a linear equation such as y = 2x + 1. A linear inequality arises when the equal sign is replaced with an inequality symbol, creating expressions such as y < 2x + 1 or y > 2x + 1.

These inequalities define regions in the coordinate plane, setting constraints such as 'less than' (<), 'greater than' (>), 'less than or equal to' (≤), or 'greater than or equal to' (≥). Solving a linear inequality involves finding all the coordinate points that make the inequality true.

For the given exercise, each line of the triangle represents a boundary where the inequality changes. For example, the inequality y ≤ 0 indicates that the region of interest is at or below the x-axis. Thus, expressing the system of inequalities for a triangle requires using these symbols to delineate where the region lies in relation to each bounding line.

Coordinate Geometry

The study of coordinate geometry, also known as analytic geometry, involves the analysis of geometric shapes in the coordinate plane using algebraic equations. The coordinate plane is a two-dimensional surface formed by the intersection of a vertical y-axis and a horizontal x-axis. Each point in this plane is defined by a pair of numerical coordinates representing its position along these two axes.

In the context of the given problem, we are looking at a triangle with vertices at (0,0), (6,0), and (1,5). To describe this triangle using coordinate geometry, we find the equations of the three lines that form the edges of the triangle. These lines are then used to define the area of the triangle - any point within this area must satisfy the equations of all three boundaries.

Visualizing Vertices

When graphing, plot the vertices on the coordinate plane and connect them with straight lines. These lines are then analyzed to establish the equations that represent them. The equations dictate the precise linear path through coordinates, allowing us to express the constraints or boundaries of a shape, such as our triangle.

Graphing Systems of Inequalities

Learning to graph a system of inequalities is essential when we need a visual representation of all the solutions to a set of inequalities. Each inequality divides the coordinate plane into two regions: one that satisfies the inequality (solution region) and one that does not.

To graph a system, follow these basic steps:

• Graph each inequality separately. Begin with the equality part (imagine the inequality sign is an equal sign).
• Choose a test point not on the line to determine which side of the line is included in the solution set.
• Shade the region of the plane that satisfies the inequality. If the inequality is strict (< or >), use a dashed line for the boundary. If it's non-strict (≤ or ≥), use a solid line.
• The solution region for the system is where the shaded areas of all inequalities overlap.

For the triangle in the exercise, you would graph each line with their respective inequality, and the area of overlap that satisfies all three inequalities represents the region within the triangle. It's a powerful visual tool that helps us solve and understand the conjunction of multiple constraints.