Section 4.1: Inequalities and Applications

Introduction to Inequalities

Inequalities are mathematical statements that compare two expressions using the symbols > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Understanding how to solve, graph, and express the solution sets of inequalities is essential in algebra and its applications.

Solving Linear Inequalities

• Solving Inequalities: The process is similar to solving equations, but with special attention when multiplying or dividing by negative numbers.

• Key Properties:

  • Adding or subtracting the same number on both sides does not change the solution set.

  • Multiplying or dividing both sides by a positive number does not change the solution set.

  • Multiplying or dividing both sides by a negative number reverses the direction of the inequality.

• Example: Solve for .

  • First, expand and simplify both sides.

  • Isolate to find the solution.

Graphing Solutions and Notation

Solutions to inequalities can be represented on a number line. The type of endpoint (open or closed) depends on whether the endpoint is included in the solution set.

• Parentheses ( ) indicate endpoints that are not included.

• Brackets [ ] indicate endpoints that are included.

Set-Builder and Interval Notation

• Set-Builder Notation: Describes the set of all numbers that satisfy a given condition, e.g., .

• Interval Notation: Uses intervals to describe all solutions, e.g., .

Examples of Notation

• : Set-builder: ; Interval:

• : Set-builder: ; Interval:

• : Set-builder: ; Interval:

• : Set-builder: ; Interval:

• : Set-builder: ; Interval:

Solving and Expressing Inequalities

• Example: Solve

  • Subtract 5 from both sides:

  • Interval notation:

  • Set-builder notation:

• Example: Solve

  • Multiply both sides by 2:

  • Interval notation:

  • Set-builder notation:

• Example: Solve

  • Divide both sides by -3 (reverse inequality):

  • Interval notation:

  • Set-builder notation:

Function Notation and Inequalities

• Example: Let and . Find all for which .

  • Set up the inequality:

  • Solve for as with any linear inequality.

  • Express the solution in both interval and set-builder notation.

Translating Words to Inequalities

• is at least → (greater than or equal to)

• is at most → (less than or equal to)

• is no more than →

• is no less than →

• greater than →

• less than →

Examples:

• I must score no less than 75 on the final:

• I have no more than $25 in my account:

Applications and Problem Solving with Inequalities

• Example: An online store offers a shipping deal: $100 per purchase. For what number of purchases is the deal better?

  • Set up the inequality:

  • Solve for to find the minimum number of purchases needed.

• Example: Lazer Line charges $65 plus $45 per hour for copier repair. Jonas was billed less than $150. How many hours was Jonas' copier worked on?

  • Set up the inequality:

  • Solve for .

Summary Table: Interval and Set-Builder Notation

The following table summarizes the correspondence between interval notation, set-builder notation, and their graphical representation on the number line:

Key Points to Remember

• Always reverse the inequality when multiplying or dividing both sides by a negative number.

• Use parentheses for open intervals and brackets for closed intervals in interval notation.

• Translate word problems into inequalities carefully, matching key phrases to the correct inequality symbol.

Practice Problems

• Solve and graph:

• Solve and graph:

• Solve and graph:

• Application: Jenn can rent a moving truck for either $99 plus per mile. For what mileages would the unlimited plan save money?

• Application: Bea can be paid (Plan A) or (Plan B) for hours. For what values of $n$ is Plan B better?