Q1. Solve the following equation: $

Background

Topic: Solving Rational Equations

This question tests your ability to solve equations that contain rational expressions (fractions with polynomials in the numerator and denominator). You need to manipulate the equation to eliminate denominators and solve for the variable.

Key Terms and Formulas:

• Rational Expression: An expression of the form where and are polynomials and .

• LCD (Least Common Denominator): The smallest expression that can be used as a common denominator for all rational terms.

Step-by-Step Guidance

1. Identify the denominators in the equation. Here, both rational expressions have as the denominator.

2. Find the LCD for all rational expressions. In this case, the LCD is .

3. Multiply both sides of the equation by the LCD to clear the denominators:

4. Simplify each term after multiplying by the LCD. This will eliminate the denominators and leave you with a polynomial equation.

5. Combine like terms and set up the resulting equation for solving. At this point, you should have a linear or quadratic equation to solve for .

Try solving on your own before revealing the answer!

Q2. Solve the equation. Begin by writing the equation with positive exponents only: $

Background

Topic: Solving Equations with Negative Exponents

This question tests your ability to rewrite expressions with negative exponents as rational expressions, and then solve the resulting equation.

Key Terms and Formulas:

• Negative Exponent Rule:

• Rational Expression: An expression involving fractions with variables in the denominator.

Step-by-Step Guidance

1. Rewrite each term with negative exponents as a rational expression:

   ,

2. Substitute these into the equation to get:

3. Find the LCD for all terms. The LCD here is .

4. Multiply both sides of the equation by the LCD to clear denominators:

5. Simplify each term after multiplying by the LCD. This will give you a quadratic equation in .

Try solving on your own before revealing the answer!