Polynomial and Rational Inequalities: Interval Notation, Set-Builder Notation, and Solution Methods

Section 1.9 Polynomial and Rational Inequalities

Key Concepts and Skills

This section covers essential techniques for solving polynomial and rational inequalities, including factoring polynomials and expressing solution sets using various notations.

• Factoring Trinomials with a Leading Coefficient Equal to 1

• Factoring Trinomials with a Leading Coefficient Not Equal to 1

• Factoring Polynomials by Grouping

• Solving Higher-Order Polynomial Equations

Writing Sets Using Set-Builder and Interval Notation

When solving inequalities, it is important to express solution sets clearly. The three main methods are:

• Interval Notation: Uses parentheses and brackets to describe intervals of real numbers.

• Set-Builder Notation: Uses a variable and a condition to describe the set.

• Number Line Graph: Visually represents the solution set.

Types of Intervals and Their Notations

The following table summarizes the main types of intervals and their corresponding notations:

Type of Interval and Graph	Interval Notation	Set-Builder Notation
Open interval	(a, b)	{x | a < x < b}
Closed interval	[a, b]	{x | a ≤ x ≤ b}
Half-open interval (left closed, right open)	[a, b)	{x | a ≤ x < b}
Half-open interval (left open, right closed)	(a, b]	{x | a < x ≤ b}
Open infinite interval (greater than a)	(a, ∞)	{x | x > a}
Open infinite interval (less than b)	(−∞, b)	{x | x < b}
Closed infinite interval (greater than or equal to a)	[a, ∞)	{x | x ≥ a}
Closed infinite interval (less than or equal to b)	(−∞, b]	{x | x ≤ b}

Solving Linear Inequalities in One Variable

Linear inequalities in one variable can be solved using algebraic techniques similar to those used for equations, with special attention to the direction of the inequality symbol.

• Definition: A linear inequality in one variable is an inequality that can be written in the form $ax + b < c$, $ax + b > c$, $ax + b eq c$, $ax + b ext{ (≤ or ≥) } c$, where $a$, $b$, and $c$ are real numbers and $a eq 0$.

• Switching the Inequality Symbol: When multiplying or dividing both sides of a linear inequality by a negative number, the direction of the inequality symbol must be reversed.

Example: Solving $2x ext{ – } 3 ext{ ≤ } 5$

• Step 1: Add 3 to both sides: $2x ext{ ≤ } 8$

• Step 2: Divide both sides by 2: $x ext{ ≤ } 4$

• Solution Set: All real numbers $x$ such that $x ext{ ≤ } 4$

• Interval Notation: $(-, 4]$

• Set-Builder Notation: ${x | x ext{ ≤ } 4}$

• Number Line: Shade all values to the left of 4, including 4.

Methods for Describing Solutions to Inequalities

There are three standard methods for expressing the solution set of an inequality:

1. Graphing the solution on a number line

2. Writing the solution in set-builder notation

3. Writing the solution in interval notation

Compound Inequalities

Compound inequalities involve two inequalities joined by "and" or "or". The solution set may be the intersection or union of the individual solution sets.

• Intersection ("and"): The solution set consists of values that satisfy both inequalities.

• Union ("or"): The solution set consists of values that satisfy at least one of the inequalities.

Example: $1 ext{ ≤ } x ext{ ≤ } 5$

• Interval Notation: $[1, 5]$

• Set-Builder Notation: ${x | 1 ext{ ≤ } x ext{ ≤ } 5}$

• Number Line: Shade between 1 and 5, including both endpoints.

Factoring Polynomials for Inequalities

Factoring is a key step in solving polynomial inequalities. The main methods include:

• Factoring trinomials with leading coefficient 1: $x^2 + bx + c$

• Factoring trinomials with leading coefficient not equal to 1: $ax^2 + bx + c$

• Factoring by grouping: Used for polynomials with four or more terms.

Example: Factoring $x^2 + 5x + 6$

• Find two numbers that multiply to 6 and add to 5: 2 and 4.

• Factor: $x^2 + 5x + 6 = (x + 2)(x + 3)$

Summary Table: Interval Types

The following table summarizes the main interval types and their notations:

Type	Interval Notation	Set-Builder Notation
Open	(a, b)	{x | a < x < b}
Closed	[a, b]	{x | a ≤ x ≤ b}
Half-open	[a, b) or (a, b]	{x | a ≤ x < b} or {x | a < x ≤ b}
Infinite	(a, ∞), [a, ∞), (−∞, b), (−∞, b]	{x | x > a}, {x | x ≥ a}, {x | x < b}, {x | x ≤ b}

Additional info: These notes are based on Section 1.9 of a College Algebra textbook and are suitable for exam preparation on inequalities and interval notation.