Lecture on Solving Rational Equations

Learning Objectives

• Distinguish rational equations
• Solve rational equations

Definition of Rational Equations

• A rational equation is an equation containing one or more rational expressions.
• A rational expression is a ratio of two polynomials.
• Examples:
  • ( \frac{1}{x} = \frac{1}{5 - x} )
  • ( \frac{x + 4}{x} = -5 )
  • ( \frac{x^2}{x + 1} = \frac{1}{x + 1} )
• All examples contain rational expressions.

Steps to Solve Rational Equations

Step 1: Solve for Value of ( x )

• Use algebraic expressions to isolate ( x ).

Step 2: Check the Solution

• Verify if the obtained solution is true.
• Extraneous Solution: A false solution that does not satisfy the equation.

Step 3: Decide on True Solutions

• True solutions are values of ( x ) that satisfy the equation.

Example 1

• Equation: ( \frac{x}{5} + \frac{1}{4} = \frac{x}{2} )

Solving Step-by-Step

1. Combine fractions on the left:
   • LCD of 5 and 4 is 20, so rewrite: ( \frac{4x + 5}{20} = \frac{x}{2} )
2. Cross multiply:
   • ( 2(4x + 5) = 20x )
3. Simplify:
   • Distribute: ( 8x + 10 = 20x )
4. Isolate ( x ):
   • ( 10 = 12x )
   • ( x = \frac{5}{6} )

Checking the Solution

• Substitute ( x = \frac{5}{6} ) into the equation.
• Verify both sides are equal.
• Conclusion: ( x = \frac{5}{6} ) is a true solution.

Example 2

• Equation: ( \frac{y + 3}{y - 1} = \frac{4}{y - 1} )

Solving Step-by-Step

1. Notice same denominators, multiply both sides by ( y - 1 ):
   • Simplified equation: ( y + 3 = 4 )
2. Solve for ( y ):
   • ( y = 1 )

Checking the Solution

• Substitute ( y = 1 ) into the equation.
• Both sides yield ( \frac{4}{0} ), which is undefined.
• Conclusion: ( y = 1 ) is an extraneous solution, thus no true solution.

Key Takeaways

• Rational equations can have true or extraneous solutions.
• Ensure solutions do not make any denominator zero.
• Always verify solutions by substitution to check validity.