Q1. Simplify the expression: \( \frac{1 + 7i}{8 + 2i} \)

Background

Topic: Complex Numbers (Division)

This question tests your ability to divide complex numbers and write the result in standard form \( a + bi \).

Key Terms and Formulas

• Complex number: A number in the form \( a + bi \), where \( i \) is the imaginary unit (\( i^2 = -1 \)).

• Standard form: \( a + bi \).

• To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

• Conjugate: If denominator is \( 8 + 2i \), its conjugate is \( 8 - 2i \).

Step-by-Step Guidance

1. Write the expression: \( \frac{1 + 7i}{8 + 2i} \).

2. Identify the conjugate of the denominator: \( 8 - 2i \).

3. Multiply both numerator and denominator by the conjugate: \( \frac{1 + 7i}{8 + 2i} \times \frac{8 - 2i}{8 - 2i} \).

4. Expand the numerator and denominator using distributive property (FOIL method).

5. Combine like terms and simplify, but stop before the final calculation.

Try solving on your own before revealing the answer!

Q2. Solve the rational inequality: \( \frac{x - 4}{x + 5} \leq 0 \)

Background

Topic: Rational Inequalities

This question tests your ability to solve rational inequalities and express the solution set in interval notation.

Key Terms and Formulas

• Rational inequality: An inequality involving a ratio of polynomials.

• Critical points: Values where numerator or denominator is zero.

• Interval notation: A way to express the solution set.

Step-by-Step Guidance

1. Set numerator and denominator equal to zero to find critical points: \( x - 4 = 0 \) and \( x + 5 = 0 \).

2. List the critical points: \( x = 4 \) and \( x = -5 \).

3. Divide the real number line into intervals based on these points: \( (-\infty, -5) \), \( (-5, 4) \), \( (4, \infty) \).

4. Test each interval by picking a value from each and substituting into the original inequality.

5. Determine which intervals satisfy the inequality, but stop before writing the final solution set.

Try solving on your own before revealing the answer!