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Magazine Chapter 1: Problem 16
Sketch the region given by the set. (a) \(\\{(x, y) | y<3\\}\) (b) \(\\{(x, y) | x=-4\\}\)
Short Answer
Expert verified
(a) Below the line \( y=3 \); (b) Line at \( x=-4 \).
Step by step solution
01
Understand the Set Notation
The notation \(\{(x, y) | y<3\}\) describes all the points \((x, y)\) on a graph where the y-coordinate is less than 3. Similarly, the notation \(\{(x, y) | x=-4\}\) describes all points where the x-coordinate is exactly -4.
02
Sketching the Region for \( y
On a Cartesian plane, the inequality \( y<3 \) includes all points below the line \( y=3 \). First, draw a horizontal dashed line at \( y=3 \) (since the line itself is not included). Then, shade the entire region below this line to represent all points where \( y<3 \).
03
Sketching the Line for \( x=-4 \)
The equation \( x=-4 \) represents a vertical line on the Cartesian plane where the x-coordinate is always -4, regardless of the y-coordinate. Draw this line as a solid line since it includes all points on \( x=-4 \).
04
Analyzing the Intersection of the Regions
Observe that the sketches from Steps 2 and 3 do not overlap in any specific manner other than composing part of the overall coordinate plane. Each sketch is independent: one describes all points below \( y=3 \), and the other describes all points along \( x=-4 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Set Notation
In mathematics and, more specifically, in coordinate geometry, set notation is a powerful tool for defining a collection of points that fulfill certain conditions. Set notation is typically presented in curly braces, indicating the set of all points
- For example, the set \[ \{(x, y) | y < 3 \} \] represents all pairs of coordinates on the Cartesian plane where the y-value is less than 3.
- Similarly, \[ \{(x, y) | x = -4 \} \] indicates every point where the x-coordinate is strictly -4, regardless of what the y-coordinate is.
Cartesian Plane
The Cartesian plane, named after the French mathematician Ren茅 Descartes, is a two-dimensional plane marked by a horizontal axis called the x-axis and a vertical axis called the y-axis. This system facilitates organizing and interpreting spatial information and relationships between different coordinates.
- The x-axis extends horizontally, where positive values move to the right and negative values move to the left from the origin point \((0, 0)\).
- The y-axis extends vertically, with positive values going upwards and negative values going downwards.
Sketching Inequalities
Graphing inequalities on the Cartesian plane offers a visual representation of all points that satisfy a given condition.
- To sketch the inequality \( y < 3 \), begin by drawing a horizontal dashed line at \((y = 3)\). The dashed line indicates that points along \( y = 3 \) are not included in the solution set, as the inequality is strictly "less than."
- Shade the region below the dashed line to represent every point where the y-coordinate is less than 3. This shading visually emphasizes the prospective values of y that satisfy the inequality.
Vertical and Horizontal Lines
Vertical and horizontal lines play a special role on the Cartesian plane, serving as crucial references and elements in plotting equations or determining regions.
- A horizontal line, such as \( y = b \), runs parallel to the x-axis. Its defining feature is that it has a constant y-coordinate for every x-value, forming a straight and level line.
- Meanwhile, a vertical line, as in \( x = a \), runs parallel to the y-axis with a consistent x-coordinate for all points along it, forming a straight up-and-down line.
