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College Algebra: Inequalities and Interval Notation

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  • How do you represent the solution to x > 3 in interval notation?
    The solution is represented as \((3, \infty)\).
  • How do you represent the solution to x ≤ -2 in interval notation?
    The solution is represented as \((-\infty, -2]\).
  • What symbols are used for open and closed intervals in interval notation?
    Parentheses ( ) denote open intervals (endpoints not included), and brackets [ ] denote closed intervals (endpoints included).
  • How do you write infinite intervals in interval notation?
    Use the infinity symbol \(\infty\) with a parenthesis, e.g., \((a, \infty)\).
  • What is a compound inequality?
    A compound inequality has two parts joined by 'AND' or 'OR'.
  • How do you solve a compound inequality?
    Solve each part separately, then find the union (OR) or intersection (AND) of the solutions.
  • What does the word 'OR' represent in compound inequalities?
    'OR' represents the union of solution sets, symbolized by \(\cup\).
  • What does the word 'AND' represent in compound inequalities?
    'AND' represents the intersection of solution sets, symbolized by \(\cap\).
  • How do you solve and graph the inequality 4 ≤ x ≤ 10?
    The solution is all x such that x is between 4 and 10, inclusive, written as \([4, 10]\).
  • How do you solve and graph the inequality x < 6 or x ≥ 10?
    The solution is the union of x less than 6 and x greater than or equal to 10, written as \((-\infty, 6) \cup [10, \infty)\).
  • What is the solution set for the inequality |x| < 3?
    The solution is all x between -3 and 3, written as \((-3, 3)\).
  • How do you interpret the inequality |x| > 3?
    The solution is x less than -3 or x greater than 3, written as \((-\infty, -3) \cup (3, \infty)\).
  • What interval notation represents the intersection of [1, 3] and (2, 6)?
    The intersection is (2, 3], written as \((2, 3]\).
  • What interval notation represents the union of [1, 3] and (2, 6)?
    The union is [1, 6), written as \([1, 6)\).
  • How do you solve the linear inequality -2x - 3 < 41?
    Add 3 to both sides: -2x < 44, then divide by -2 (reverse inequality): x > -22.
  • How do you solve the linear inequality 2 - 3x ≥ 5?
    Subtract 2: -3x ≥ 3, divide by -3 (reverse inequality): x ≤ -1.
  • What is the solution to the compound inequality 4x - 1 < 11 and 2x - 3 ≥ -1?
    Solve each: 4x < 12 → x < 3; 2x ≥ 2 → x ≥ 1; solution is x in [1, 3).
  • How do you solve the compound inequality 5 < 2x - 5 < 11?
    Add 5: 10 < 2x < 16; divide by 2: 5 < x < 8; solution is (5, 8).
  • What is the solution to the absolute value inequality |2x + 6| ≤ 8?
    Solve -8 ≤ 2x + 6 ≤ 8; subtract 6: -14 ≤ 2x ≤ 2; divide by 2: -7 ≤ x ≤ 1.
  • What is the solution to the absolute value inequality |4 - x| < 5?
    Solve -5 < 4 - x < 5; subtract 4: -9 < -x < 1; multiply by -1 (reverse): -1 < x < 9.