Solving Rational Inequalities

Introduction

• Focus of the video: Solving rational inequalities.
• Previous topic covered: Rational equations.

Example Problem

• Solve the inequality:
  [ \frac{x^2 - 3x - 4}{x^2 - 8x + 16} < 0 ]

Steps to Solve

1. General Form

   • Ensure the inequality is in the form:
     • Left side: polynomial expressions.
     • Right side: 0.

2. Find Critical Values

   • Factor the numerator and denominator:
     • Numerator: ( x^2 - 3x - 4 = (x - 4)(x + 1) )
     • Denominator: ( x^2 - 8x + 16 = (x - 4)(x - 4) = (x - 4)^2 )
   • Set each factor equal to zero to find critical values:
     • ( x - 4 = 0 ) → ( x = 4 )
     • ( x + 1 = 0 ) → ( x = -1 )

3. Identify Regions

   • Critical values divide the number line into three regions:
     • Region 1: ( (-\infty, -1) )
     • Region 2: ( (-1, 4) )
     • Region 3: ( (4, \infty) )

4. Test Each Region

   • Choose test points from each region:
     • Region 1: Use ( x = -2 )
       • Substitute into inequality: [ \frac{(-2 - 4)(-2 + 1)}{(-2 - 4)^2} ] → evaluate to check if less than zero.
     • Region 2: Use ( x = 0 )
       • Substitute into inequality: [ \frac{(0 - 4)(0 + 1)}{(0 - 4)^2} ] → evaluate to check.
     • Region 3: Use ( x = 5 )
       • Substitute into inequality: [ \frac{(5 - 4)(5 + 1)}{(5 - 4)^2} ] → evaluate to check.

5. Evaluate Results

   • Region 1: False (not included)
   • Region 2: True (included)
   • Region 3: False (not included)

Critical Values Inclusion

• Check whether critical values are included:
  • For ( x = -1 ): false (open circle)
  • For ( x = 4 ): false (open circle)

Final Answer

• Solution in set builder notation:
  [ x \mid -1 < x < 4 ]

Conclusion

• Encouragement to like and subscribe for more content.
• Reminder: Understanding of rational inequalities is crucial.