Rational Equations

Definition

Rational equations are algebraic equations that contain one or more fractions. These equations involve the division of polynomials, where the numerator and denominator are both polynomial expressions. Solving rational equations is a fundamental skill in pre-algebra, as it allows students to work with and manipulate fractional expressions within an equation.

5 Must Know Facts For Your Next Test

1. Rational equations can be solved by using techniques such as cross-multiplying, finding the least common denominator, and factoring the numerator and denominator.
2. The solution to a rational equation may result in extraneous solutions, which do not satisfy the original equation and must be discarded.
3. Rational equations can model real-world situations involving rates, proportions, and other fractional relationships.
4. Solving rational equations often requires students to manipulate and simplify complex fractions, which reinforces their understanding of fraction operations.
5. Rational equations can be classified as linear, quadratic, or higher-degree, depending on the degree of the numerator and denominator polynomials.

Review Questions

• Explain the process of solving a rational equation, including the steps involved and the importance of identifying and discarding any extraneous solutions.
  • To solve a rational equation, the first step is to clear the fractions by cross-multiplying the numerator and denominator terms on both sides of the equation. This creates a new polynomial equation without any fractions. Next, the student must simplify the equation by combining like terms and factoring, if possible. Finally, the student must solve the resulting polynomial equation and check the solutions to ensure they satisfy the original rational equation. It is important to identify and discard any extraneous solutions, as they do not represent valid solutions to the original rational equation.
• Describe how rational equations can be used to model real-world situations, and provide an example of a practical application.
  • Rational equations can be used to model a variety of real-world situations that involve fractional relationships, such as rates, proportions, and inverse variations. For example, a rational equation could be used to represent the relationship between the speed of a vehicle, the distance traveled, and the time taken. The equation could be used to solve for an unknown variable, such as the time required to travel a certain distance at a given speed. By understanding how to solve rational equations, students can apply this knowledge to solve practical problems that involve fractional relationships in various contexts, such as finance, science, and engineering.
• Analyze the similarities and differences between solving rational equations and solving other types of algebraic equations, such as linear or quadratic equations. Discuss the unique challenges and considerations involved in working with rational expressions.
  • Solving rational equations shares some similarities with solving other types of algebraic equations, such as the need to isolate the variable of interest and use techniques like factoring and simplifying. However, rational equations present unique challenges due to the presence of fractions in the equation. Working with rational expressions requires a deeper understanding of fraction operations, including finding the least common denominator and simplifying complex fractions. Additionally, rational equations can result in extraneous solutions that do not satisfy the original equation, which must be identified and discarded. The process of solving rational equations often involves multiple steps and requires careful attention to the algebraic manipulations involved. Compared to linear or quadratic equations, rational equations can be more complex and require a higher level of algebraic reasoning and problem-solving skills.

Related terms

Polynomial:

A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers.

Fraction:

A fraction is a representation of a part of a whole, expressed as a numerator divided by a denominator.

Least Common Denominator (LCD):

The least common denominator is the smallest positive integer that is divisible by all the denominators in a set of fractions, used to simplify rational expressions.