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Magazine Chapter 2: Problem 42
Write an inequality that describes all points in the lower half-plane below the \(x\) -axis.
Short Answer
Expert verified
The inequality is y < 0.
Step by step solution
01
Understanding the Coordinate Plane
The coordinate plane is a two-dimensional plane with a horizontal axis (x-axis) and a vertical axis (y-axis). Points in the plane are described by pairs (x, y). Here, the task is to describe the points that lie below the x-axis.
02
Define the x-Axis in Terms of y
The x-axis is defined by all points where the y-coordinate is zero, i.e., y = 0. This forms the boundary between the upper and lower halves of the plane.
03
Describe Points Below the x-Axis
Points located in the lower half-plane have y-coordinates that are less than zero because they are below the x-axis boundary.
04
Write the Inequality
The inequality that encompasses all points in the lower half-plane is given by the inequality y < 0. This states that any point (x, y) where y is less than zero lies below the x-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Coordinate Plane
In mathematics, the coordinate plane is a flat surface defined by two intersecting lines: the horizontal axis known as the x-axis, and the vertical axis known as the y-axis. These axes divide the plane into four quadrants. Every point on this plane can be described using a pair of numbers \(x, y\), which are even terms as coordinates.
- The first number (x-coordinate) tells us how far left or right the point is from the origin (0,0).
- The second number (y-coordinate) shows how far up or down the point is from the origin.
Introduction to x-axis and y-axis
The x-axis and y-axis serve as reference lines for the coordinate plane. Understanding their roles is crucial.
The x-axis runs horizontally across the plane. It is the line where the value of \(y\) is always zero. This axis acts as a dividing line between points above and below, clearly distinguishing the positive part of the plane from the negative part.
The y-axis, in contrast, runs vertically. Along this axis, the value of \(x\) is always zero. It determines the left side from the right side within each quadrant.
The x-axis runs horizontally across the plane. It is the line where the value of \(y\) is always zero. This axis acts as a dividing line between points above and below, clearly distinguishing the positive part of the plane from the negative part.
The y-axis, in contrast, runs vertically. Along this axis, the value of \(x\) is always zero. It determines the left side from the right side within each quadrant.
- The positive direction of the x-axis is to the right, while the negative is to the left.
- The upwards point is positive for the y-axis, while downwards is negative.
Graphing Inequalities on the Coordinate Plane
Graphing inequalities involves a few key steps. First, we must know the equation of the boundary line, which is often presented as an equal sign (\(=\)). But because we deal with inequalities, we instead use signs like less than (\(<\)) or more than (\(>\)), which indicate the half-plane where the solution exists.
To graph an inequality:
To graph an inequality:
- Start by sketching the boundary line, replacing the inequality sign with an equal sign for that step.
- If the inequality is "less than" or "greater than," draw a dashed line; it indicates the points on the line are not part of the solution.
- For "less than or equal to" or "greater than or equal to," use a solid line.
- Once the line is drawn, determine which side of the line fulfills the inequality and shade that region.
Understanding Lower Half-Plane Inequality
The term "lower half-plane" refers to the region below the x-axis. Understanding how to define this region with an inequality is essential for clear representation in mathematics.
The x-axis is the demarcation line; all points with a \(y\)-value less than zero are in the lower half-plane.
The x-axis is the demarcation line; all points with a \(y\)-value less than zero are in the lower half-plane.
- The inequality \(y < 0\) succinctly captures all points \((x, y)\) that are located beneath the x-axis.
- This inequality is a simple yet effective way to represent the entire lower half of the coordinate plane visually and algebraically.
