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

Background

Topic: Complex Numbers

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

Key Terms and Formulas

• Complex number: An expression of the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) 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.

Key formula:

Conjugate of \( 8 + 2i \) is \( 8 - 2i \).

Multiply numerator and denominator by the conjugate:

Step-by-Step Guidance

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

2. Multiply both the numerator and denominator by the conjugate:

   in the numerator and in the denominator.

3. Expand the numerator using the distributive property:

4. Expand the denominator using the difference of squares:

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 is testing your ability to solve rational inequalities and express the solution set in interval notation.

Key Terms and Formulas

• Rational inequality: An inequality involving a rational expression (a fraction with polynomials in numerator and denominator).

• Interval notation: A way to describe the set of solutions using intervals.

Key formula:

To solve \( \frac{x - 4}{x + 5} \leq 0 \), find where the expression is zero or negative.

Step-by-Step Guidance

1. Identify the critical points by setting the numerator and denominator equal to zero:

   and

2. Find the values of \( x \) where the expression is undefined or equals zero.

3. Set up intervals based on the critical points and test values in each interval to determine where the expression is less than or equal to zero.

4. Remember to exclude values that make the denominator zero from the solution set.

Try solving on your own before revealing the answer!