The Main Idea

• Interval Notation:
  • A concise way to represent ranges of real numbers
  • Uses brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints
  • Incorporates infinity symbols for unbounded intervals
• Inequalities:
  • Mathematical statements comparing two expressions
  • Use symbols like <, >, ≤, ≥ to show relationships between values
• Set-Builder Notation:
  • Another way to represent sets of numbers
  • Example: [latex]{x \mid 0 \leq x \leq 4}[/latex] represents all [latex]x[/latex] between [latex]0[/latex] and [latex]4[/latex]
• Number Line Representation:
  • Visual method to display intervals and inequalities
  • Uses solid dots for inclusive endpoints, open circles for exclusive endpoints

Key Points

1. Interval Types:
   • Open interval: [latex](a, b)[/latex]
   • Closed interval: [latex][a, b][/latex]
   • Half-open intervals: [latex][a, b)[/latex] or [latex](a, b][/latex]
   • Infinite intervals: [latex](a, \infty)[/latex] or [latex](-\infty, b)[/latex]
2. Union of Intervals:
   • Represented by the ∪ symbol
   • Example: [latex](-\infty, a) \cup (b, \infty)[/latex] represents all numbers less than [latex]a[/latex] or greater than [latex]b[/latex]
3. Translating Between Notations:
   • Inequality → Interval Notation → Number Line → Words
   • Each representation conveys the same information in a different format

The Main Idea

• Addition Property:
  • If [latex]a < b[/latex], then [latex]a + c < b + c[/latex]
  • Adding or subtracting the same number on both sides preserves the inequality
• Multiplication Property:
  • If [latex]a < b[/latex] and [latex]c > 0[/latex], then [latex]ac < bc[/latex]
  • If [latex]a < b[/latex] and [latex]c < 0[/latex], then [latex]ac > bc[/latex]
  • Multiplying or dividing by a positive number preserves the inequality
  • Multiplying or dividing by a negative number reverses the inequality
• These properties apply to [latex]\leq[/latex], [latex]>[/latex], and [latex]\ge[/latex] as well

Key Points

1. Solving Inequalities:
   • Similar to solving equations, but with attention to inequality direction
   • Isolate the variable on one side of the inequality
   • Perform the same operations on both sides
2. Reversing Inequalities:
   • When multiplying or dividing by a negative number, reverse the inequality sign
   • This is crucial for maintaining the correct solution set
3. Solution Sets:
   • Often expressed in interval notation
   • Represent all real numbers that satisfy the inequality

The Main Idea

• Definition:
  • A compound inequality combines two inequalities in one statement
  • Can be connected by “and” (conjunction) or “or” (disjunction)
• Types of Compound Inequalities:
  • “And” Compound Inequality: Both conditions must be true (e.g., [latex]a < x < b[/latex])
    • Represent a range of values between two bounds
    • Solution is the intersection of the two individual inequalities
  • “Or” Compound Inequality: At least one condition must be true (e.g., [latex]x < a \text{ or } x > b[/latex])
    • Represent values satisfying at least one of the conditions
    • Solution is the union of the two individual inequalities
• Solving Methods:
  • Separate into two inequalities and solve individually
  • Solve the compound inequality intact, operating on all parts simultaneously
  • Apply the same operation to all parts of the inequality
  • Pay attention to inequality direction when multiplying or dividing by negatives
  • Combine results for the final solution