Difference of Rational Function, Rational Equation, and Rational Inequalities

What is a Rational Expression?

• Rational Expression: Can be written as a ratio of two polynomials.
  • No negative or fractional exponents.
  • Variables are not inside a radical.
• Examples:
  • 2/x
  • (x^2 + 2x + 3)/(x + 1)
  • 5/(x - 3)

Analysis of Algebraic Expressions

• x^2 + 3x + 2 / (x + 4): Rational expression because both numerator and denominator are polynomials.
• 1/(3x^2): Rational expression because the numerator and denominator are polynomials.
• x^2 + 4x - 3 / 2: Rational expression.
• √(x+1) / (x^3 - 1): Not a rational expression because the numerator is inside a radical.
• x^(-2) - 5 / (x^3 - 1): Not a rational expression because the exponent in the numerator is negative.
• 1/x + 2/(x-2): Rational expression when simplified.

Difference of Rational Equation, Inequality, and Function

Rational Equation

• Equation using a rational expression.
• Example: 5/x - 3/2x = 1/5
• Note: Has the "=" symbol.

Rational Inequality

• Inequality using a rational expression.
• Example: 5/x - 3 ≤ 2/x
• Note: Has inequality symbols: ">", "<", "≤", "≥".

Rational Function

• Function in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0.
• Example: f(x) = (x^2 + 2x + 3)/(x + 1)
• Note: In the form f(x) or y.

Practice Exercises

1. 2 + x/(x - 1) = 8: Rational Equation
2. x > √(x + 2): None (not rational)
3. f(x) = 6 - (x + 3)/(x^2 - 5): Rational Function
4. 2x ≥ 7/(x + 4): Rational Inequality
5. x/2 = 4/x + 9x^3: Rational Equation

Analysis:

• Check if the expression is rational.
• Check the symbols to identify if it's an equation, inequality, or function.

Conclusion

• Rational Equation: Has the "=" symbol.
• Rational Inequality: Has inequality symbols (">", "<", "≤", "≥").
• Rational Function: In form of f(x) or y.

Remember: A rational equation/inequality is solved for all x values that satisfy the condition, while a rational function shows the relationship between two variables like x and y.