In mathematics, numbers are categorized into different sets:

• Integers: Whole numbers, including negatives, zero, and positives:
  $$ { \dots, -3, -2, -1, 0, 1, 2, 3, \dots } $$
• Rational Numbers: Numbers that can be written as a fraction of two integers, such as:
  $$ \frac{3}{4}, \quad -2, \quad 0.75, \quad 5 $$
• Irrational Numbers: Numbers that cannot be written as fractions, such as:
  $$ \pi, \quad \sqrt{2}, \quad e $$
• Real Numbers: The full set of rational and irrational numbers combined.

The term "rational expression" does not mean that $x$ is rational. In fact, $x$ is usually assumed to be in the real numbers unless stated otherwise. "Rational expression" just means a ratio of polynomials, not an expression restricted to rational numbers.

What is a Rational Expression?

A rational expression is a fraction where both the numerator and denominator are polynomials:

where $P(x)$ and $Q(x)$ are polynomials, and $Q(x)\ne 0$.

Rational expressions behave similarly to numerical fractions, meaning they follow the same rules for simplification, multiplication, division, addition, and subtraction.

Operations with Rational Expressions

1. Simplifying: Factor the numerator and denominator and cancel common factors.
2. Multiplication/Division: Multiply numerators together and denominators together.
3. Addition/Subtraction: Find a common denominator before combining terms.

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Practice Problems

1. Simplify the expression:

(A)

(B)

(C)

(D)

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2. Multiply and simplify:

(A)

(B)

(C)

(D)

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3. Divide and simplify:

(A)

(B)

(C)

(D)

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4. Add and simplify:

(A)

(B)

(C)

(D)

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5. Solve for $x$ in the equation:

(A)

(B)

(C)

(D)