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Chapter 8: Problem 69

Find a system of inequalities to describe the given region. Points inside the square that has vertices \((2,2),(-2,2)\) \((-2,-2),\) and \((2,-2)\).

Short Answer

Expert verified
The system of inequalities is \( -2 \leq x \leq 2 \) and \( -2 \leq y \leq 2 \).

Step by step solution

01

- Identify the boundaries of the square

Observe the vertices \( (2,2), (-2,2), (-2,-2), (2,-2) \). These points create a square with sides parallel to the x and y axes.
02

- Determine the inequalities for the x-coordinates

The x-coordinates of the vertices are \(2 \) and \(-2\). This indicates that the x-coordinates of points inside the square must satisfy \(-2 \leq x \leq 2 \).
03

- Determine the inequalities for the y-coordinates

The y-coordinates of the vertices are \(2 \) and \(-2\). This indicates that the y-coordinates of points inside the square must satisfy \(-2 \leq y \leq 2 \).
04

- Combine the inequalities

The system of inequalities describing the points inside the square is the combination of the two findings: \( -2 \leq x \leq 2 \) and \( -2 \leq y \leq 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertices
Vertices are the corner points of a geometric shape. In our problem, the vertices of the square are \(2,2\), \(-2,2\), \(-2,-2\), and \(2,-2\). These points are crucial as they define the overall size and shape of the square. Each pair of adjacent vertices connects to form the sides of the square. Identifying these points is the first step in understanding the boundaries and constraints of the problem. Vertices help us determine the range of x and y values that describe the region inside the square.
x-coordinates
The x-coordinates of the vertices in our square are \(2\) and \(-2\). These values tell us the horizontal extent of the square. Points within the square must have x-coordinates that fall within the range defined by these two values. Hence, the inequality that describes this constraint is \(-2 \leq x \leq 2\). This means that any point within the square has an x-coordinate between \(-2\) and \(2\), inclusive. Understanding the x-coordinates helps narrow down where points can lie horizontally in the x-axis.
y-coordinates
Similarly, the y-coordinates of the vertices are \(2\) and \(-2\). These values determine the vertical extent of the square. Consequently, for a point to be inside the square, its y-coordinate must satisfy the inequality \(-2 \leq y \leq 2\). This inequality indicates that any point within the boundary of the square has its y-coordinate ranging from \(-2\) to \(2\), inclusive. By understanding the y-coordinates, we map out where points can lie vertically on the y-axis.
Boundaries
Boundaries define the limits of a geometric shape. In our problem, the boundaries of the square are derived from the x and y inequalities. For the given vertices \(2,2\), \(-2,2\), \(-2,-2\), and \(2,-2\), the boundaries can be described with the system of inequalities: \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\). These inequalities combine to describe the region inside the square. Any point that satisfies both inequalities lies within the boundaries of the square, making it easier to identify all valid points in the region.

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