Lecture on Inequalities

Key Concepts

• Inequalities: Mathematical statements that compare expressions using inequality symbols.
• Symbols Used in Inequalities:
  • <: Less than
  • >: Greater than
  • ≤: Less than or equal to
  • ≥: Greater than or equal to
  • These symbols may also be represented by phrases like "is fewer than," "is at most," "is at least," etc.

Writing Inequalities

• Example A:
  • Sentence: "A number w minus 3.5 is less than or equal to -2."
  • Inequality: w - 3.5 ≤ -2
• Example B:
  • Sentence: "3 is less than a number n plus 5."
  • Inequality: 3 < n + 5
• Example C:
  • Sentence: "0 is greater than or equal to twice a number x plus 1."
  • Inequality: 0 ≥ 2x + 1

Testing Solutions

• Plugging Values:
  • Plug a number into an inequality to see if it makes a true statement.
  • Example: Test if -4 is a solution.
    • -4 + 8 < -3 is false (4 is not less than -3).
    • -4.5 x -4 > -21 is true (18 is greater than -21).

Graphing Inequalities

• Number Line Representation:

  • Open Circle: Used when a number is not included in the solution set.
  • Closed Circle: Used when a number is included.
  • Arrows: Indicate the direction of the inequality.

• Graphing Examples:

  • Example A: y ≤ -3
    • Use closed circle at -3; arrow pointing left.
  • Example B: 2 < x
    • Open circle at 2; arrow pointing right.
  • Example C: x > 0
    • Open circle at 0; arrow pointing right.

Real-World Examples

• Amusement Park Ride Restrictions:
  • Ride A:
    • Graph shows a closed circle at 48, pointing right.
    • Inequality: h ≥ 48
  • Ride B:
    • Graph shows an open circle at 52, pointing left.
    • Inequality: h < 52

Conclusion

• Mastery of inequalities involves understanding symbols, writing and interpreting inequalities, testing solutions, and graphing them effectively. These skills are valuable for solving real-world problems.