Problem 94 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{2 x}{1-x}+\frac{6 x}{x-1} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{2x}{1-x} + \frac{6x}{x-1}$$. Answer: The simplified expression is $$\frac{-4x}{1-x}.$$

Step 1: Identify the given expressions

We are given two rational expressions: $$ \frac{2x}{1-x} \text{ and } \frac{6x}{x-1} $$ Our task is to add or subtract these rational expressions.

Step 2: Find the common denominator

In order to add or subtract rational expressions, we first need to find a common denominator. In this case, we notice that the denominators are the same, but with opposite signs. We can rewrite the second denominator with the same sign as the first one, like this: $$ \frac{2x}{1-x} + \frac{6x}{-(x-1)} $$ Now the common denominator is \((1-x)\).

Step 3: Rewrite the fractions with the common denominator

We need to rewrite both fractions with the common denominator \((1-x)\). The first fraction is already in this form, so we only need to adjust the second fraction. We have: $$ \frac{2x}{1-x} + \frac{-6x}{1-x} $$

Step 4: Combine the numerators

Now that we have both expressions with the same denominator, we can combine the numerators and keep the same denominator. We get: $$ \frac{2x - 6x}{1-x} $$

Step 5: Simplify the expression

The last step is to simplify the expression. In the numerator, we can combine the terms with \(x\) to get: $$ \frac{-4x}{1-x} $$ This is the simplified form of the given sum of rational expressions.

Key Concepts

Common Denominator

Understanding the concept of a common denominator is essential when dealing with rational expressions, especially during addition or subtraction. A denominator is the bottom part of a fraction, and when fractions have different denominators, it is often necessary to find a common denominator before performing any operations on them.

In the context of our example, the rational expressions \(\frac{2x}{1-x}\) and \(\frac{6x}{x-1}\) initially seem to have different denominators. However, by observing that \(x-1\) is simply \(1-x\) multiplied by \(-1\), we realize that the denominators are closely related. Thus, a common denominator for both expressions is \(1-x\), which allows us to combine the expressions into a single fraction. Always remember to look for such relationships between denominators; this skill is crucial in simplifying complex rational expressions.

Simplify Rational Expressions

The process of simplifying rational expressions involves reducing them to a simpler form while maintaining their original value. To illustrate, let's take our combined expression \(\frac{2x - 6x}{1-x}\). Simplification is primarily carried out in the numerator since the common denominator remains unchanged.

We combine like terms in the numerator by subtracting \(6x\) from \(2x\), which results in \(\frac{-4x}{1-x}\). This form is more eloquent and manageable as it shows the relationship between the numerator and the denominator clearly. To simplify rational expressions, always look for opportunities to combine like terms, factor out common factors, or cancel out terms that appear both in the numerator and denominator.

Algebraic Fractions

Algebraic fractions are simply fractions that contain variables, such as \(x\), in their numerators, denominators, or both. They operate under the same rules as regular fractions, but with the added complexity of variables.

When you encounter algebraic fractions, handle them with care, ensuring that you apply algebraic rules correctly. It's also crucial to check for potential restrictions on the variables. For example, in the expression \(\frac{-4x}{1-x}\), it's important to note that \(x\) cannot be equal to 1 since division by zero is undefined. Always confirm the values of variables for which the algebraic fraction is valid, as this affects how you deal with the expression in equations or inequalities.

Rational Expressions Subtraction

Subtracting rational expressions follows the same principle as adding them, requiring a common denominator to proceed. The difficulty often lies in manipulating the expressions to achieve that common denominator without altering the value of the fractions.

In our example, we added \(\frac{2x}{1-x}\) and \(\frac{6x}{x-1}\), but if we wanted to subtract them, we would use the same approach to obtain a common denominator before subtracting the numerators. When tackling subtraction problems, always ensure that you distribute the subtraction through the numerator if necessary and simplify the result just as you would with addition problems. Remember, subtracting rational expressions correctly often involves meticulous attention to signs; a common pitfall is losing track of negative signs, which can completely alter the outcome.