R.1 Sets and the Real Number Line

Miller & Gerken, College Algebra 2e — Pages 2-17

Learning Objectives

Identify subsets of the real numbers
Use interval notation and graph inequalities
Evaluate absolute value expressions
Find the distance between two points on a number line

1. Sets of Numbers

Number Systems

Natural Numbers \(\mathbb{N}\): Counting numbers: \(1, 2, 3, \ldots\)
Whole Numbers \(\mathbb{W}\): Natural numbers plus zero: \(0, 1, 2, 3, \ldots\)
Integers \(\mathbb{Z}\): Whole numbers plus negatives: \(\ldots, -2, -1, 0, 1, 2, \ldots\)
Rational Numbers \(\mathbb{Q}\): Fractions \(\frac{p}{q}\) where \(q \neq 0\). Examples: \(\frac{1}{2}, -3, 0.75\)
Irrational Numbers: Cannot be written as fraction. Examples: \(\sqrt{2}, \pi, e\)
Real Numbers \(\mathbb{R}\): All rational and irrational numbers combined

Every integer is rational: \(5 = \frac{5}{1}\)

The number sets form a nested hierarchy:

\[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\]

2. Interval Notation

Notation	Set-Builder	Description
\((a, b)\)	\(\{x \mid a < x < b\}\)	Open interval — both endpoints excluded
\([a, b]\)	\(\{x \mid a \leq x \leq b\}\)	Closed interval — both endpoints included
\([a, b)\)	\(\{x \mid a \leq x < b\}\)	Half-open — closed at \(a\), open at \(b\)
\((a, \infty)\)	\(\{x \mid x > a\}\)	Unbounded right — open at \(a\)
\((-\infty, b]\)	\(\{x \mid x \leq b\}\)	Unbounded left — closed at \(b\)

Notation

Set-Builder

Description

\((a, b)\)

\(\{x \mid a < x < b\}\)

Open interval — both endpoints excluded

\([a, b]\)

\(\{x \mid a \leq x \leq b\}\)

Closed interval — both endpoints included

\([a, b)\)

\(\{x \mid a \leq x < b\}\)

Half-open — closed at \(a\), open at \(b\)

\((a, \infty)\)

\(\{x \mid x > a\}\)

Unbounded right — open at \(a\)

\((-\infty, b]\)

\(\{x \mid x \leq b\}\)

Unbounded left — closed at \(b\)

Parenthesis \(( )\) = endpoint NOT included (open circle)
Bracket \(\[ ]\) = endpoint IS included (closed circle)
Infinity \(\infty\) always gets parenthesis (can’t "reach" infinity)

3. Absolute Value

Definition

\[|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\]

Absolute value = distance from zero. Always non-negative.

Examples

\[|7| = 7\]

\[|-7| = -(-7) = 7\]

\[|0| = 0\]

4. Distance Between Two Points

Formula

\[d(a, b) = |b - a|\]

Example

Distance between \(-3\) and \(5\):

\[d = |5 - (-3)| = |5 + 3| = |8| = 8\]

Practice Problems

Problem 1 (p. 15, #12)

Classify the number: \(-\sqrt{49}\)

My work:

\[-\sqrt{49} = -7\]

\(-7\) is an integer, therefore also rational and real.

Answer: Integer, Rational, Real

Problem 2 (p. 15, #23)

Simplify: \(|3 - 7| - |2 - 9|\)

My work:

\[\begin{align} |3 - 7| - |2 - 9| &= |-4| - |-7| \\ &= 4 - 7 \\ &= -3 \end{align}\]

Answer: \(-3\)

Problem 3 (p. 16, #47)

Write in interval notation: \(x \geq -2\)

My work:

\(x\) can be \(-2\) or anything greater, going to infinity.

Closed bracket at \(-2\) (included)
Parenthesis at \(\infty\) (never included)

Answer: \([-2, \infty)\)

Problem 4 (p. 16, #63)

Find the distance between: \(-8\) and \(3\)

My work:

\[d = |b - a| = |3 - (-8)| = |3 + 8| = |11| = 11\]

Answer: \(11\)

Section Summary

Real numbers include rationals and irrationals: \(\mathbb{R} = \mathbb{Q} \cup \{\text{irrationals}\}\)
Interval notation: \(\[ ]\) = included, \(( )\) = excluded
\(|x|\) = distance from zero, always \(\geq 0\)
Distance between \(a\) and \(b\): \(d = |b - a|\)

Study Notes

Started this section to refresh fundamentals. The absolute value definition with cases is important — it explains why \(|-7| = -(-7) = 7\).

Need to practice interval notation more — keep mixing up when to use brackets vs parentheses.

Next: R.2 Integer Exponents