Equations and Inequalities

Linear Inequalities and Absolute Value Inequalities

This section covers the fundamental concepts and techniques for solving linear inequalities and absolute value inequalities, including interval notation, intersections and unions of intervals, and graphical representations. These skills are essential for understanding solution sets and their properties in College Algebra.

Objectives

• Use interval notation to describe solution sets.

• Find intersections and unions of intervals.

• Solve linear inequalities in one variable.

• Recognize inequalities with no solution or all real numbers as solutions.

• Solve compound inequalities.

• Solve absolute value inequalities.

Solving an Inequality

Solving an inequality involves finding all numbers that make the inequality true. These numbers form the solution set of the inequality. The process is similar to solving equations, but special attention must be paid to the direction of the inequality symbol, especially when multiplying or dividing by negative numbers.

Interval Notation

• Open interval (a, b): The set of real numbers between, but not including, a and b.

• Closed interval [a, b]: The set of real numbers between, and including, a and b.

• Infinite intervals: - (a, \infty): Real numbers greater than a. - (-\infty, b]: Real numbers less than or equal to b.

• Parentheses indicate endpoints not included; brackets indicate endpoints included.

Example: Using Interval Notation

• Interval: [1, 3.5]

• Set-builder notation:

• Graph: A number line with a solid dot at 1 and 3.5, and shading between them.

Finding Intersections and Unions of Two Intervals

To analyze the relationship between two intervals:

1. Graph each interval on a number line.

2. Intersection: The portion common to both intervals.

3. Union: The total collection of numbers in either interval.

Example: Intersections and Unions

• Intervals: [1, 3] and (2, 6)

• Intersection: (2, 3] (numbers in both intervals)

• Union: [1, 6) (all numbers in either interval)

Solving Linear Inequalities in One Variable

A linear inequality in x can be written as:

Important rule: When multiplying or dividing both sides by a negative number, reverse the direction of the inequality symbol.

Example: Solving a Linear Inequality

• Problem: Solve

• Solution:

• Interval notation:

• Graph: Number line shaded to the left of 5, with an open circle at 5.

Recognizing Inequalities with No Solution or All Real Numbers as Solutions

• If the inequality is always true (e.g., ), the solution set is all real numbers: .

• If the inequality is never true (e.g., ), the solution set is empty: .

Example: Compound Inequality

• Problem: Solve

• Interval notation:

• Set-builder notation:

• Graph: Number line with a solid dot at -1 and an open dot at 4, shading between.

Solving Absolute Value Inequalities

Let be an algebraic expression and a positive number:

• : The solution is

• : The solution is or

These rules also apply for and inequalities.

Example: Solving an Absolute Value Inequality

• Problem: Solve

• Rewrite:

• Solution:

• Interval notation:

• Graph: Number line shaded between -3 and 7, with open circles at both endpoints.