Chapter 1: Variables, Constants, Plotting Points, and Inequalities – Foundations for Introductory Statistics

Chapter 1: Performing Operations and Evaluating Expressions

1.1 Variables, Constants, Plotting Points, and Inequalities

This section introduces foundational mathematical concepts essential for statistics, including variables, constants, and the representation of numbers and inequalities. Understanding these concepts is crucial for interpreting data and constructing mathematical models in statistics.

Variables

• Definition: A variable is a symbol that represents a quantity that can change or vary.

• Example: Let h be the height (in feet) of a specific child. Since height can change as time passes, h is a variable. If h = 4, the child's height is 4 feet.

• Application: Variables are used to represent unknown or changing quantities in mathematical expressions and statistical models.

Examples of Variables in Context

• Speed: Let s be a car's speed (in miles per hour). If s = 60, the car's speed is 60 mph.

• Population: Let n be the number of people (in millions) who work from home at least half the time. If n = 3.7 in 2017, then 3.7 million people worked from home at least half the time that year.

• Time: Let t be the number of years since 2015. If t = 5, then the year is 2020.

Constants

• Definition: A constant is a symbol that represents a specific number—a quantity that does not vary.

• Example: In the area formula for a rectangle, if the area is always 12 square inches, then A = 12 is a constant.

Comparing Constants and Variables

• In the context of a rectangle with area 12 square inches:

  • Variables: W (width) and L (length) can vary as long as their product is 12.

  • Constant: A (area) is constant at 12 square inches in this scenario.

Types of Numbers

• Counting Numbers (Natural Numbers): 1, 2, 3, 4, ...

• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...

• Rational Numbers: Numbers that can be written as $\frac{n}{d}$, where n and d are integers and d ≠ 0. Their decimal representations either terminate or repeat.

• Irrational Numbers: Numbers that cannot be written as $\frac{n}{d}$, and their decimal representations neither terminate nor repeat (e.g., $\pi$, $\sqrt{2}$).

• Real Numbers: All numbers that can be represented on the number line, including both rational and irrational numbers.

Visualizing Numbers on the Number Line

• Each point on the number line corresponds to a real number.

• Numbers increase from left to right; the distance between consecutive integers is 1 unit.

Data

• Definition: Data are quantities or categories that describe people, animals, or things.

• Examples: Heights of people (e.g., 64, 71, 75 inches), genres of music (e.g., pop, rock, country).

Graphing Data

• Data can be represented visually using number lines or coordinate systems.

• Example: The total amounts (in billions of dollars) of loans for a company over several years can be plotted on a number line to observe trends.

• When plotting, label the axis with the variable and its units.

The Coordinate System

• Ordered Pair: A pair of numbers (x, y) representing a point in the plane.

• Axes: Two perpendicular number lines (x-axis and y-axis) intersecting at the origin (0, 0).

• Quadrants: The plane is divided into four quadrants by the axes.

• Plotting Points: The first number in the ordered pair is the x-coordinate (horizontal), and the second is the y-coordinate (vertical).

• Example: The point (3, 4) is located 3 units to the right of the origin and 4 units up.

Inequalities

• Symbols and Meanings:

  • < : is less than

  • ≤ : is less than or equal to

  • > : is greater than

  • ≥ : is greater than or equal to

• Example: $3 < 6$ is true because 3 is less than 6.

• Graphing Inequalities: Use a number line, shading the region representing the solution set. Open circles indicate endpoints not included; closed circles indicate included endpoints.

Interval Notation

• Used to describe sets of numbers between two endpoints.

• Examples:

  • $(a, b)$: All numbers greater than a and less than b (endpoints not included).

  • $[a, b]$: All numbers from a to b, including both endpoints.

  • $(a, b]$: All numbers greater than a and up to and including b.

  • $[a, b)$: All numbers from a up to but not including b.

Compound Inequalities

• Express situations where a variable must satisfy two inequalities simultaneously.

• Example: $3 \leq x \leq 7$ means $x$ is between 3 and 7, inclusive.

• Graph by shading the region between the endpoints and using closed circles if endpoints are included.

Table: Inequality Symbols and Their Meanings

Symbol	Meaning	Example
<	is less than	2 < 5
≤	is less than or equal to	4 ≤ 7
>	is greater than	9 > 2
≥	is greater than or equal to	8 ≥ 3

Examples of Describing and Graphing Inequalities

• Wind Speed: For a Category 5 hurricane, wind speed $s \geq 157$ mph. Interval notation: $[157, \infty)$.

• Baggage Weight: For a checked bag not overweight, $0 < w \leq 50$ pounds. Interval notation: $(0, 50]$.

• Man's Weight: $173 < w < 177$ pounds. Interval notation: $(173, 177)$.

Guidelines for Writing Good Mathematical Responses

• Create examples to illustrate concepts or procedures.

• Use complete sentences and correct terminology to describe key ideas.

• Describe the general concept or procedure, not just the example.

• Point out similarities and differences with related concepts.

• Explain the benefits of understanding the concept or procedure.

• Clarify why certain steps are permissible and discuss common misunderstandings.

Summary: The Meaning of a Variable

• A variable is a symbol that stands for an amount that can vary.

• A constant is a symbol that stands for an amount that does not vary.

• Variables allow concise description and manipulation of changing quantities in mathematics and statistics.