An algebraic fraction is a rational expression that represents a division of two algebraic expressions. It consists of a numerator and a denominator, where the numerator is the dividend and the denominator is the divisor. Algebraic fractions are used to solve rational equations, which are equations that contain one or more rational expressions.
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Algebraic fractions can be used to model and solve real-world problems involving rates, ratios, and proportions.
The value of an algebraic fraction is determined by the values assigned to the variables in the numerator and denominator.
Multiplying or dividing algebraic fractions involves applying the rules of fraction arithmetic, such as cross-multiplying and reciprocals.
Adding or subtracting algebraic fractions requires finding a common denominator and then applying the rules of fraction arithmetic.
Solving rational equations involves isolating the rational expression and then using algebraic techniques to find the solution(s).
Review Questions
Explain the process of simplifying an algebraic fraction and provide an example.
To simplify an algebraic fraction, you need to factor the numerator and denominator, then cancel any common factors between them. For example, consider the fraction $\frac{2x^2 - 8x}{x^2 - 4x}$. First, factor the numerator and denominator: $\frac{2x(x - 4)}{x(x - 4)}$. Then, cancel the common factor of $(x - 4)$ in the numerator and denominator, resulting in the simplified fraction $\frac{2x}{x}$, which can be further simplified to $2$.
Describe how to add or subtract two algebraic fractions with different denominators, and provide an example.
To add or subtract algebraic fractions with different denominators, you first need to find a common denominator. This is done by finding the least common multiple (LCM) of the denominators. Once you have the common denominator, you can then convert the fractions to equivalent fractions with the common denominator and perform the addition or subtraction. For example, to add the fractions $\frac{2x}{x + 1}$ and $\frac{3}{x + 1}$, you would first find the LCM of the denominators, which is $(x + 1)$. Then, you would convert the fractions to $\frac{2x}{x + 1}$ and $\frac{3(x + 1)}{x + 1}$, respectively, and add them together to get $\frac{2x + 3(x + 1)}{x + 1}$, which simplifies to $\frac{5x + 3}{x + 1}$.
Explain the process of solving a rational equation and discuss the importance of this skill in the context of 7.4 Solve Rational Equations.
Solving a rational equation involves isolating the rational expression and then using algebraic techniques to find the solution(s). This typically involves cross-multiplying, factoring, and finding common denominators. The ability to solve rational equations is crucial in the context of 7.4 Solve Rational Equations because it allows you to model and solve real-world problems that can be represented using rational expressions, such as those involving rates, ratios, and proportions. By mastering the skills of simplifying, adding, subtracting, and solving algebraic fractions, you will be better equipped to tackle the types of rational equations that may appear on your test.
A rational expression is a fraction where both the numerator and denominator are polynomials.
Simplifying Algebraic Fractions: The process of reducing an algebraic fraction to its simplest form by factoring the numerator and denominator and canceling common factors.