Problem 28 - Algebra A Combined Function

Question

Find the \(L C D\) for each list of rational expressions. $$ \frac{2 x+5}{3 x-7}, \frac{5}{7-3 x} $$

Step-by-Step Solution

Answer

The LCD is \(3x - 7\).

Step 1: Identify Denominators

The first step is to identify the denominators of each rational expression. In this case, the denominators are \(3x - 7\) and \(7 - 3x\).

Step 2: Simplify the Denominators

Notice that the second denominator \(7 - 3x\) can be rewritten as \(-1(3x - 7)\) by factoring out \(-1\). This means \(3x - 7\) and \(7 - 3x\) are equivalent up to a negative sign.

Step 3: Establish the Common Denominator

Since both denominators can be represented by \(3x - 7\) up to a sign factor, the least common denominator (LCD) of the two expressions is simply \(3x - 7\).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator contains a polynomial. This can make them a bit trickier than regular fractions. Understanding them is important to solving problems in algebra, as they appear frequently. They behave much like regular fractions:

• A rational expression is undefined if its denominator is zero, just like a fraction is undefined for zero divisors.
• To simplify or solve rational expressions, finding a common denominator is often needed.

In the given exercise, you are asked to find the least common denominator (LCD) for two rational expressions:\[\frac{2x+5}{3x-7} \quad \text{and} \quad \frac{5}{7-3x}\]To do this, examining the denominators is key to simplifying the process.

Factoring

Factoring is breaking down a complex expression into simpler, multiplied factors. It’s a critical skill for finding the least common denominators.For the rational expressions given, you first need to simplify to a form that allows you to easily spot equivalent expressions. In this context:1. Start with the denominator \(7 - 3x\) and factor out \(-1\), which reveals it as \(-1(3x - 7)\).2. Now, notice that both denominators, \(3x - 7\) and \(-1(3x - 7)\), are essentially the same, differing only by a negative sign.Factoring helps in making such discoveries, ensuring that you don’t miss equivalent forms of expressions. Being comfortable with this technique is vital in handling and simplifying complex algebraic expressions.

Equivalent Expressions

Equivalent expressions are different algebraic expressions that simplify to the same value or have the same mathematical value when solved.

• In rational expressions, finding equivalent forms can simplify finding common denominators or solving equations.
• An equivalent expression can be derived by factoring, applying properties of equality, or manipulating the expression through other algebraic means.

In the exercise, the expressions \(3x - 7\) and \(7 - 3x\) are found to be equivalent up to a negative sign through factoring:\[7 - 3x = -1(3x - 7)\]This means that, although they appear different, their equivalence assists in identifying the least common denominator as simply being \(3x - 7\). Knowing how to recognize and work with equivalent expressions can greatly simplify the process of solving rational expressions.