Problem 87 - Algebra for College Students

Question

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{2 x+4}{6}=\frac{?}{3}$$

Step-by-Step Solution

Answer

The missing expression is \( x + 2 \).

Step 1 - Simplify the Left-Hand Side Expression

First, simplify the expression on the left-hand side. The given expression is \( \frac{2x + 4}{6} \). Factor out 2 from the numerator: \( \frac{2(x + 2)}{6} \).

Step 2 - Simplify the Fraction

Next, simplify the fraction \( \frac{2(x + 2)}{6} \). Notice that \(6 \) can also be written as \( 3 \cdot 2 \). Therefore, \( \frac{2(x + 2)}{6} = \frac{2(x + 2)}{2 \cdot 3} \). We can cancel the common factor of 2 in the numerator and the denominator: \( \frac{x + 2}{3} \).

Step 3 - Set the Equivalent Expressions

Because \( \frac{2(x + 2)}{6} \) simplifies to \( \frac{x + 2}{3} \), we can see that the two expressions are equivalent: \( \frac{2x + 4}{6} = \frac{x + 2}{3} \). Therefore, the missing expression on the right-hand side is \( x + 2 \).

Key Concepts

factoring

Factoring is a useful algebraic technique that simplifies complex expressions. It involves breaking down a polynomial into simpler terms, called factors, that when multiplied together give the original polynomial.
For instance, consider the expression \(2x + 4\). By factoring out the common factor (which is 2 in this case), we rewrite it as \(2(x + 2)\).
This step is essential in simplifying rational expressions or solving equations, as it makes further operations more manageable.
Factoring serves as a foundational tool in algebra and is often the first step in transforming an expression or equation into a simpler form.

common factors

Identifying common factors between the numerator and the denominator helps in simplifying rational expressions.
For the example \(\frac{2(x + 2)}{6}\), we notice that '2' is a common factor in both the numerator and the denominator (since 6 can be written as \(2 \cdot 3\)).
Once we cancel out the common factor of '2', we get the simplified form \(\frac{x + 2}{3}\).
Recognizing and canceling common factors not only simplifies calculations but also aids in understanding the relationship between different parts of an expression.

equivalent expressions

Two expressions are considered equivalent if they yield the same value for any substitution of variables. In mathematical terms, these expressions simplify to the same form.
From our exercise, starting with \(\frac{2x + 4}{6}\) and simplifying it to \(\frac{x + 2}{3}\), we confirm that these two are indeed equivalent. Despite appearing different initially, the process of simplification reveals their equality.
Understanding equivalent expressions is crucial for solving algebraic equations, verifying solutions, and transforming expressions into more convenient forms for problem-solving.