Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Find the missing factor and state any domain restrictions necessary to make the two fractions equivalent.\(\left.\frac{5}{2 x}=\frac{(5}{6 x^{2}}\right)\)

Step-by-Step Solution

Answer

The missing factor is \(x = 1/3\) and the domain restriction is \(x \neq 0\).

Step 1: Cross Multiply

Cross multiply the fractions to get the equivalence relationship, this results in \(5 * 6x^{2} = 5 * 2x\).

Step 2: Simplifying the Equation

Simplify both sides of the equation. First, conduct the multiplication operation: \( 30x^{2} = 10x \).

Step 3: Solve for the missing factor

Since both sides of the equation are multiple of x and x cannot be zero (division by zero is undefined), divide the equation by x on both sides to find that missing factor: 30x = 10 which gives \(x = 10/30\) or \(x = 1/3\)

Step 4: Finding Restrictions

The fraction can be undefined if the divisor, x, is zero. Therefore, the restriction is \(x \neq 0\) since division by zero is undefined.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations involving two fractions set to equal each other. When you have a fraction equation like \( \frac{a}{b} = \frac{c}{d} \), you can cross multiply to find the relationship between the numerators and the denominators. Doing so involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the two products equal to each other, hence the term 'cross' multiplication.

For example, in the problem \(\frac{5}{2x} = \frac{(5}{6x^2}\), we apply cross multiplication to get \(5 \cdot 6x^2 = 5 \cdot 2x\). This method helps simplify the process of finding equivalent fractions or solving for variables. Remember, cross multiplication only works if both fractions are equal and represent proportions. It is often used in problems involving ratios and proportions and is a fundamental skill in algebra.

Solving Algebraic Fractions

Handling Algebraic Fractions:
When dealing with algebraic fractions, the steps involved often include simplifying expressions and solving for variables. After cross multiplying, you may encounter a simplified equation like \(30x^2 = 10x\). The goal now is to isolate the variable you're solving for. If a variable appears on both sides of the equation, you can divide through by the variable (provided it isn't zero, in which case it would be undefined), just as we divided by \(x\) to find \(x = 1/3\) in the example.

Keep in mind that simplification includes canceling out common factors and combining like terms when appropriate. In more complex scenarios, algebraic fractions may require additional techniques such as factoring, applying the distributive property or finding a common denominator before further simplification can occur. Throughout the process, each step needs to preserve the equivalency to maintain a correct solution.

Domain Restrictions

Understanding Domain Restrictions:
When working with algebraic fractions, it's crucial to note that certain values might make an expression undefined. These are known as domain restrictions, which are values that aren't allowed in a set of possible inputs (known as the domain). In the context of fractions, the restriction often comes from a denominator that cannot be zero because division by zero is undefined.

In our example equation \(\frac{5}{2x} = \frac{(5}{6x^2}\), there is an implicit understanding that \(x \eq 0\). This is because if \(x\) were zero, we'd have division by zero, which is not permitted in math. The statement of domain restrictions like \(x \eq 0\) ensures that all the operations within the equation are valid. Whenever solving algebraic fractions, always check for and state domain restrictions to avoid invalid solutions and to be precise about the conditions under which the equation holds true.