Problem 5 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Find the sum or difference. \(\frac{9}{x+1}-\frac{2 x}{x+1}\)

Step-by-Step Solution

Answer

The sum or difference of the given rational expressions is \(\frac{9 - 2x}{x + 1}\).

Step 1: Identifying the problem

First, it is recognized that both fractions have the same denominator \(x+1\). This means they can be combined.

Step 2: Applying arithmetic adjustment

Apply the rule of combining two fractions with the same denominator: \(\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}\). The arithmetic adjustment with \(a = 9\) and \(b = 2x\) results in \(\frac{9 - 2x}{x + 1}\).

Key Concepts

Arithmetic of Rational Expressions

When dealing with arithmetic of rational expressions, we're essentially working with fractions whose numerators and denominators are algebraic expressions. As in the case of numerical fractions, the same general rules apply. For instance, to add or subtract rational expressions, they must have a common denominator. Upon securing a common denominator, the numerators are combined using addition or subtraction.

By understanding this principle, you can easily tackle problems like combining \( \frac{9}{x+1} - \frac{2x}{x+1} \) by simply focusing on the numerators. Think of the denominator as the shared ground floor of a building, and the numerators as different floors that are being restructured. Once aligned ('combined'), these floors can be merged or divided, which in algebraic terms translates to adding or subtracting the numerators.

Common Denominators

Finding common denominators is a crucial step in the process of adding or subtracting rational expressions. A denominator is the 'common ground' of fractions, and when they are the same, it simplifies the process of combining the fractions. It's akin to synchronizing the beat of two different songs so that they can be played together without a dissonant clash.

When you have an exercise such as \( \frac{9}{x+1} - \frac{2x}{x+1} \), where both fractions already share a common denominator, you're in luck because you can proceed to the next step without additional work. However, if they do not have the same denominator, you would first need to find the least common denominator (LCD) before you could merge the expressions.

Simplifying Algebraic Fractions

The process of simplifying algebraic fractions is like cleaning up an equation to make it neater and more understandable. It involves reducing the expressions to their simplest form. This can be done by combining like terms, factoring, and canceling out common factors between the numerator and denominator.

Consider the fraction obtained from the given exercise, \( \frac{9 - 2x}{x+1} \). Even though this fraction is already quite simple, you might encounter more complex situations where additional steps, such as factoring polynomials, are necessary. The guiding principle is to look for any opportunities to reduce the fraction to its most basic form without altering the value of the expression.