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Chapter 1: Problem 39

Exer. \(39-40:\) Describe the set of all points \(P(x, y)\) in a coordinate plane that satisfy the given condition. (a) \(x=-2\) (b) \(y=3\) (c) \(x \geq 0\) (d) \(x y>0\) (e) \(y<0\) (f) \(|x| \leq 2\) and \(|y| \leq 1\)

Short Answer

Expert verified
(a) Vertical line at \(x = -2\); (b) Horizontal line at \(y = 3\); (c) Right half-plane; (d) First & third quadrants; (e) Lower half-plane; (f) Rectangle centered at origin.

Step by step solution

01

Understanding vertical lines

Set (a): The condition is \(x = -2\). This represents a vertical line in the coordinate plane at \(x = -2\). All points \((x, y)\) on this line have \(x\) equal to -2, and they can have any real value for \(y\).
02

Understanding horizontal lines

Set (b): The condition is \(y = 3\). This represents a horizontal line in the coordinate plane at \(y = 3\). All points \((x, y)\) on this line have \(y\) equal to 3 and can have any real value for \(x\).
03

Understanding half-planes

Set (c): The condition is \(x \geq 0\). This represents the right half-plane including the \(y\)-axis. All points \((x, y)\) that satisfy this inequality have \(x\) greater than or equal to 0 and any value for \(y\).
04

Understanding quadrants

Set (d): The condition is \(xy > 0\). This means that \(x\) and \(y\) must have the same sign, hence \((x, y)\) lies in either the first (\(x > 0, y > 0\)) or third quadrants (\(x < 0, y < 0\)) of the coordinate plane.
05

Understanding half-planes continued

Set (e): The condition is \(y < 0\). This represents the half-plane below the \(x\)-axis. All points \((x, y)\) that satisfy this inequality have \(y\) less than 0 and any value for \(x\).
06

Understanding rectangles

Set (f): The conditions are \(|x| \leq 2\) and \(|y| \leq 1\). This represents a rectangle centered at the origin with vertices at \((2, 1)\), \((2, -1)\), \((-2, 1)\), and \((-2, -1)\). All points \((x, y)\) within this rectangle satisfy both absolute inequalities.

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Key Concepts

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

Vertical Line
Imagine the coordinate plane as a giant piece of grid paper. A vertical line is a straight line that goes up and down. When you have an equation like \(x = -2\), it represents a vertical line that crosses the \(x\)-axis at -2. This line includes all points where the \(x\)-coordinate is -2, regardless of what the \(y\)-coordinate is.
  • Vertical lines have an undefined slope because they rise infinitely without running left or right.
  • All points on a vertical line have the same \(x\)-value, and this consistency is what gives them their vertical nature.
In geometry, vertical lines are important for defining boundaries, such as walls or barriers that run straight up and down.
Horizontal Line
Now, let's flip our thinking to side-to-side lines, called horizontal lines. A horizontal line like \(y = 3\) cuts across the \(y\)-axis at 3 and runs left and right.
  • Each point on this line will have a \(y\)-coordinate of 3 with any \(x\)-coordinate.
  • Unlike vertical lines, horizontal lines have a slope of zero because there is no rise in the line, only a run side to side.
Horizontal lines are often used to signify a surface or horizon, flat and even across distances.
Half-Planes
Half-planes are like splitting the coordinate plane into sections. When you have an inequality like \(x \geq 0\), it means you're dealing with half the plane, specifically the right-hand side of the plane, including the \(y\)-axis.
  • Here, all points have an \(x\)-coordinate that's zero or more, stretching infinitely to the right.
  • Another example is \(y < 0\), which represents the half-plane below the \(x\)-axis. In this scenario, every point has a \(y\)-coordinate that's less than 0.
Half-planes are incredibly useful in mathematics and real-world scenarios where limits or boundaries must be established within a plane.
Quadrants
The coordinate plane is divided into four regions called quadrants. These are based on the signs of \(x\) and \(y\).
  • The first quadrant is where both \(x\) and \(y\) are positive \((x > 0, y > 0)\).
  • The third quadrant is where both \(x\) and \(y\) are negative \((x < 0, y < 0)\).
With \(xy > 0\), the product of \(x\) and \(y\) implies they have the same sign, placing the point in either the first or third quadrants. Quadrants are crucial for locating points and understanding symmetry in coordinate geometry.
Rectangles
In the world of geometry, rectangles are four-sided figures with right-angle corners. When you see conditions like \(|x| \leq 2\) and \(|y| \leq 1\), you're looking at defining a specific rectangle on the coordinate plane. Here, the absolute values constrain \(x\) and \(y\) to a particular range.
  • The rectangle is centered at the origin, with its width stretching from \(-2\) to 2 on the \(x\)-axis and height from \(-1\) to 1 on the \(y\)-axis.
  • Points inside this rectangle satisfy both conditions, meaning they fit within the defined area.
This rectangle helps in understanding bounded regions that can be used for various calculations and graphical representations.

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