Intersections, Unions, and Compound Inequalities: Study Notes for Intermediate Algebra

Section 4.2: Intersections, Unions, and Compound Inequalities

Introduction

This section explores compound inequalities, focusing on conjunctions (intersections) and disjunctions (unions), and their graphical representations. It also covers the domain of functions using set-builder and interval notation. These concepts are fundamental in Intermediate Algebra for understanding how solution sets interact and how to describe them precisely.

Conjunctions (Intersections)

Definition and Properties

A conjunction is a compound sentence formed by joining two or more inequalities or statements with the word and. The solution set of a conjunction is the intersection of the solution sets of the individual statements, meaning only values that satisfy all conditions are included.

• Intersection of Sets: The intersection of sets A and B, written as $A \cap B$, is the set of all elements common to both A and B.

• Conjunction of Inequalities: For two inequalities joined by "and," the solution set is the overlap (intersection) of their individual solution sets.

• Set-builder Notation: Used to describe the set of solutions, e.g., $\{x \mid -2 < x \leq 0\}$.

• Interval Notation: Used to describe the range of solutions, e.g., $(-2, 0]$.

Example: Solve and graph $-2 < x$ and $x \leq 0$.

• Set-builder notation: $\{x \mid -2 < x \leq 0\}$

• Interval notation: $(-2, 0]$

Solving Compound Inequalities

To solve compound inequalities joined by "and," solve each inequality separately and find the overlap.

• Example: $3 \leq 2x - 1 < 7$

• Break into two inequalities: $3 \leq 2x - 1$ and $2x - 1 < 7$

• Solve each:

  • $3 \leq 2x - 1 \implies 4 \leq 2x \implies 2 \leq x$

  • $2x - 1 < 7 \implies 2x < 8 \implies x < 4$

• Combined solution: $2 \leq x < 4$

• Set-builder notation: $\{x \mid 2 \leq x < 4\}$

• Interval notation: $[2, 4)$

Disjunctions (Unions)

Definition and Properties

A disjunction is a compound sentence formed by joining two or more inequalities or statements with the word or. The solution set of a disjunction is the union of the solution sets of the individual statements, meaning any value that satisfies at least one condition is included.

• Union of Sets: The union of sets A and B, written as $A \cup B$, is the set of all elements that belong to A or B (or both).

• Disjunction of Inequalities: For two inequalities joined by "or," the solution set is the combined (union) of their individual solution sets.

• Set-builder Notation: Used to describe the set of solutions, e.g., $\{x \mid x < -4 \text{ or } x \geq 2\}$.

• Interval Notation: Used to describe the range of solutions, e.g., $(-\infty, -4) \cup [2, \infty)$.

Example: Solve and graph $x < -4$ or $x \geq 2$.

• Set-builder notation: $\{x \mid x < -4 \text{ or } x \geq 2\}$

• Interval notation: $(-\infty, -4) \cup [2, \infty)$

Solving Compound Inequalities with "Or"

To solve compound inequalities joined by "or," solve each inequality separately and combine their solution sets.

• Example: $x \geq 3$ or $x > -4$

• Solution: All $x > -4$ (since $x \geq 3$ is included in $x > -4$)

• Set-builder notation: $\{x \mid x > -4\}$

• Interval notation: $(-4, \infty)$

Set-builder and Interval Notation

Definitions

• Set-builder notation: Describes a set by stating the property that its members must satisfy, e.g., $\{x \mid x > 2\}$.

• Interval notation: Describes a set as a range of values, e.g., $(2, \infty)$.

Both notations are used to express solution sets for inequalities and domains of functions.

Domains of Functions

Definition and Determining Domains

The domain of a function is the set of all possible input values (x-values) for which the function is defined. To find the domain, identify values that make the function undefined (such as division by zero or taking the square root of a negative number).

• Example 1: $f(x) = 5x - 2$ Domain: All real numbers ($\mathbb{R}$)

• Example 2: $f(x) = \frac{2x+1}{x-3}$ Domain: $x \neq 3$ Set-builder notation: $\{x \mid x \neq 3\}$ Interval notation: $(-\infty, 3) \cup (3, \infty)$

• Example 3: $f(x) = \sqrt{8 - 2x}$ Domain: $8 - 2x \geq 0 \implies x \leq 4$ Set-builder notation: $\{x \mid x \leq 4\}$ Interval notation: $(-\infty, 4]$

Summary Table: Conjunctions vs. Disjunctions

Type	Word Used	Set Operation	Solution Set
Conjunction	and	Intersection ($A \cap B$)	Values satisfying both conditions
Disjunction	or	Union ($A \cup B$)	Values satisfying at least one condition

Things to Remember and Mistakes to Avoid

• For conjunctions, only values that satisfy all conditions are included.

• For disjunctions, values that satisfy at least one condition are included.

• Always check for values that make a function undefined when finding domains.

• Use set-builder and interval notation to clearly describe solution sets.