Problem 60 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The system of inequalities that describes the region of the given triangle are: \(y\leq-x+1\), \(y\leq x+1\), and \(y\geq 0\).

Step 1: Identify the Vertices

The vertices provided are (-1,0), (1,0), and (0,1). These vertices will form a triangle. The first task is to get the equations of the lines that pass through these vertices.

Step 2: Calculating the Equations of Lines

The equation of a line in a 2D plane is given by \(y=mx+b\), where m is the slope and b is y-intercept. Three lines are needed from the triangle: \n1. A line passing through (-1,0) and (1,0) gives \(y=0\) \n2. A line passing through (-1,0) and (0,1) gives \(y=-x+1\) \n3. A line passing through (1,0) and (0,1) gives \(y=x+1\).

Step 3: Forming the Inequalities

Since we are trying to describe the region inside the triangle, all points (x, y) that lie on or below the line \(y=-x+1\) will satisfy the inequality \(y\leq-x+1\). Similarly, all points (x, y) that lie on or below the line \(y=x+1\) will satisfy the inequality \(y\leq x+1\). And for the line \(y=0\), all points that lie on or above it will satisfy the inequality \(y\geq 0\). Hence the system of inequalities that describes the region of the triangle is: \n \(y\leq-x+1, y\leq x+1, y\geq 0\).

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and represent relationships where variables change at a constant rate. You can recognize a linear equation by its standard form, which is \(y = mx + b\), where \(m\) represents the slope, or steepness, of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In the provided exercise, we see linear equations generated by connecting the given vertices of a triangle. A key point is identifying the slope between two points. For example, the line connecting the points (-1, 0) and (1, 0) has a slope of 0, hence the equation \(y = 0\), which is a horizontal line.

When given a pair of points, you can calculate the slope by dividing the change in y by the change in x, often articulated as 'rise over run'. Understanding linear equations is essential because they serve as the building blocks for more complex functions and systems.

Graphing Inequalities

Graphing inequalities is a way of visualizing the solution set of an inequality on a coordinate plane. Unlike linear equations that show equality, inequalities deal with less than, greater than, or their 'or equal to' counterparts. Importantly, when graphing an inequality, one must remember to distinguish open dots, used to denote '<' or '>', from closed dots, representing '\(\leq\)' or '\(\geq\)'.

For instance, in the system from the exercise, the inequality \(y \leq -x + 1\) is graphed as a line, then the area below that line, representing all the possible \(y\) values that satisfy the inequality, is shaded. Similarly, the inequality \(y \geq 0\) indicates that the solution region is above the horizontal axis. A solid line is used because the '=' part of the inequality includes the points on the line itself. By understanding and graphing these inequalities, students can better visualize and solve problems involving constrained regions, such as the interior of a triangle.

The Slope-Intercept Form

The slope-intercept form \(y = mx + b\) is one of the most intuitive ways to represent linear equations because it clearly shows the slope and the y-intercept. The slope \(m\) determines how tilted the line is on a graph, and when \(m = 0\) as in our horizontal line example, we see it as a flat line parallel to the x-axis.

The y-intercept \(b\) shows where the line crosses the y-axis. This form makes it easy to graph linear equations without needing to calculate multiple points. Simply plot the y-intercept on the y-axis and then use the slope to find another point on the line.

In the example problem, the slope-intercept form aids in quickly determining the equations for the triangle's edges, demonstrating its practicality in geometric problems. It ensures that students can methodically progress from the equations of lines to the visualization of a geometric region defined by inequalities.