Sets & Inequalities

1. Sets

• A set is a well-defined collection of objects (elements). Notation: roster form {1, 2, 3}, or set-builder form {x | condition on x}.
• Operations:
  • Union A∪B: elements in A or B (or both).
  • Intersection A∩B: elements in both A and B.
  • Complement A′ (relative to a universal set U): elements in U but not in A.
  • Difference A−B: elements in A but not in B.
• Venn diagrams are used to visualize these operations.
• Subsets: A⊆B means every element of A is also in B. The empty set ∅ is a subset of every set.

Common mistakes: confusing A∩B (both) with A∪B (either); forgetting the empty set is a subset of every set including itself; when computing A′, forgetting to specify the universal set U.

2. Quadratic Inequalities

To solve ax² + bx + c > 0 (or <, ≥, ≤):

1. Find the roots of ax² + bx + c = 0 (factor or use the quadratic formula).
2. Since the parabola opens upward when a > 0, the expression is negative between the roots and positive outside the roots.
3. For a < 0, the sign pattern flips (parabola opens downward).

Example: x² − 5x + 6 < 0 factors as (x−2)(x−3) < 0. Since a=1>0, the product is negative between the roots: solution is 2 < x < 3.

Common mistakes: forgetting to flip the inequality sign when dividing by a negative number; assuming the solution is always "outside the roots" without checking whether the parabola opens up or down; writing the answer with the roots in the wrong order.

3. Rational Inequalities

For inequalities like p(x)/q(x) > 0 (or <, ≥, ≤):

1. Find the zeros of the numerator p(x) and the zeros of the denominator q(x) — these are the "critical points."
2. The denominator's zeros must always be excluded from the solution (division by zero is undefined), even if the inequality is ≥ or ≤.
3. Use a sign chart: test the sign of the whole expression in each interval formed by the critical points.

Common mistakes: including a value that makes the denominator zero in a "≥" or "≤" solution (it must always be excluded); forgetting that multiplying both sides by an expression that could be negative requires flipping the inequality — this is why sign-chart analysis is safer than cross-multiplying directly.