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Chapter 12: Problem 4

Write a system of inequalities for the indicated region. Quadrant IV, not including the \(x\) - and \(y\) -axes

Short Answer

Expert verified
x > 0 and y < 0

Step by step solution

01

Identify the boundaries

In Quadrant IV, the boundary lines are the positive x-axis and the negative y-axis. However, since the boundaries do not include the axes, we will express the inequalities as strict inequalities.
02

Define the regions for x and y

In Quadrant IV, x is positive and y is negative. Therefore, we express this as two separate inequalities: 1. x > 0 (x must be positive) 2. y < 0 (y must be negative)
03

Combine the inequalities

Combining the inequalities from Step 2 gives us the system of inequalities that describes all the points in Quadrant IV while excluding the axes.

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Key Concepts

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

Quadrants
The coordinate plane is divided into four quadrants. Each quadrant is a specific region defined by the signs of the x and y coordinates. Quadrant IV is one of these four sections.
Quadrant IV is located in the bottom right part of the coordinate plane. In this quadrant:
  • x is positive
  • y is negative
Knowing which quadrant a point belongs to helps in graphing and solving systems of inequalities. For example, if a point is in Quadrant IV, it means the point has positive x and negative y values.
Graphing Inequalities
Graphing inequalities on a coordinate plane helps to visualize solutions. For Quadrant IV, we need to graph inequalities that fall within the region where x is positive and y is negative.
When graphing inequalities:
  • Use a dashed line for inequalities like x > 0 and y < 0 because the inequalities do not include the boundary lines themselves.
  • Shade the region of the graph that satisfies the inequality.
To graph the system for Quadrant IV, start by drawing the lines x = 0 (asymptote) and y = 0 (asymptote). Then, because we have x > 0 and y < 0, shade the area to the right of the y-axis and below the x-axis. This shaded region is Quadrant IV, excluding the axes.
Coordinate Plane
The coordinate plane consists of two perpendicular lines called axes. These axes divide the plane into four quadrants.
  • The horizontal axis is called the x-axis.
  • The vertical axis is called the y-axis.
The point where the x-axis and y-axis intersect is called the origin, labeled as (0,0).
Each point on the plane is represented by a pair of values \((x,y)\) that indicate its position relative to the origin. Positive and negative values of x and y determine the point's placement within a particular quadrant. Remember:
  • Quadrant I: x is positive, y is positive
  • Quadrant II: x is negative, y is positive
  • Quadrant III: x is negative, y is negative
  • Quadrant IV: x is positive, y is negative
Understanding these concepts is crucial for solving problems involving systems of inequalities and graphing accurately.

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