Lecture Notes: Polynomial and Rational Inequalities

1. Polynomial Inequalities

• Example Problem: Solve (x^3 - 4x < 0)
  • Factorization: (x(x^2 - 4) = x(x-2)(x+2))
  • Critical Points: x = 0, 2, -2
  • Sub-intervals:
    • ((-\infty, -2))
    • ((-2, 0))
    • ((0, 2))
    • ((2, \infty))
  • Test Points:
    • (x = -3): negative
    • (x = -1): positive
    • (x = 1): negative
    • (x = 3): positive
  • Solution: ((-\infty, -2) \cup (0, 2))

2. Rational Inequalities

• Example Problem: (\frac{2x+4}{x-1} < 1)
  • Rearrange: (\frac{2x+4}{x-1} - 1 < 0)
  • Common Denominator: (\frac{2x+4 - (x-1)}{x-1} < 0)
  • Simplify: (\frac{x+5}{x-1} < 0)
  • Critical Points:
    • Numerator zero at (x = -5)
    • Denominator zero at (x = 1)
  • Sub-intervals:
    • (x < -5)
    • ((-5, 1))
    • ((1, \infty))
  • Solution:
    • Test Points:
      • (x = -6): positive
      • (x = 0): negative
      • (x = 2): positive
    • For (< 0), solution is ((-5, 1))
    • For (\leq 0), solution is (-5, 1)) since (x \neq 1).

3. Greater than or Equal Rational Inequalities

• Example Problem: (\frac{2x-1}{x^2-9} \geq 0)
  • Factorization:
    • Denominator: ((x-3)(x+3))
  • Critical Points:
    • Numerator zero at (x = \frac{1}{2})
    • Denominator zero at (x = \pm 3)
  • Sub-intervals:
    • (( -\infty, -3))
    • ((-3, \frac{1}{2}))
    • ((\frac{1}{2}, 3))
    • ((3, \infty))
  • Test Points:
    • (x = -4): negative
    • (x = 0): positive
    • (x = 1): negative
    • (x = 4): positive
  • Solution:
    • Positive intervals: ((-3, \frac{1}{2}] \cup (3, \infty))
    • Cannot include (x = \pm 3) due to zero in the denominator.

Summary

• Key Takeaway: Solving polynomial and rational inequalities involves:
  • Factoring the polynomial/rational expression
  • Identifying critical points by setting numerator and denominator to zero
  • Dividing the number line into intervals
  • Testing intervals for sign determination
  • Being cautious with inequalities involving denominators to avoid division by zero.