Problem 58 - Algebra A Combined Function

Question

Write each phrase as an expression. The reciprocal of \(x\), subtracted from the reciprocal of 5

Step-by-Step Solution

Answer

The expression is \(\frac{1}{5} - \frac{1}{x}\).

Step 1: Understand the Phrase

The phrase "The reciprocal of \(x\)" means you need to find a value that, when multiplied by \(x\), equals 1. So, the reciprocal of \(x\) is \(\frac{1}{x}\). Similarly, "the reciprocal of 5" is \(\frac{1}{5}\).

Step 2: Write the Subtraction Expression

The word "subtracted from" indicates the order of the subtraction, meaning you'll subtract the reciprocal of \(x\) from the reciprocal of 5. Therefore, the expression becomes \(\frac{1}{5} - \frac{1}{x}\).

Key Concepts

Reciprocals

In mathematics, a reciprocal refers to the number that, when multiplied by the original number, results in an answer of 1. This is a fundamental concept that frequently emerges in various mathematical operations. To find the reciprocal of a number, you simply take 1 and divide it by that number.

• For instance, the reciprocal of 5 is \left(\frac{1}{5}\right). This is because \left(5 \, \times \, \frac{1}{5} = 1\right).
• Similarly, for a variable, say \(x\), its reciprocal is \(\frac{1}{x}\).

Understanding reciprocals is crucial, especially when dealing with problems involving fractions and equations. They allow for simplification and transformation of equations that include fractions, like turning division problems into multiplication tasks, which can often be more straightforward. By grasping the concept of reciprocals, students can navigate more effectively through algebraic expressions, ensuring a solid foundation for more complex mathematical concepts.

Subtraction in Algebra

Subtraction in algebra may seem a lot like regular subtraction, but it presents its unique challenges and concepts. In algebra, subtraction is more than just taking numbers away from one another; it involves carefully dealing with expressions and their variables.

• The phrase 'subtracted from' is pivotal. For example, 'subtracting \(a\) from \(b\)' should be written as \(b - a\). Always pay attention to the order.
• Subtraction can also involve negative numbers and operations that change the sign of terms, affecting the overall outcome of the expression.

When performing subtraction in algebra, it's essential to maintain the integrity of the variables involved and respect the operations at play. This ensures precise and accurate results. It's a skill that enhances the comprehension of creating and solving expressions, especially as they become more complex.

Writing Expressions

Writing expressions is a core component of algebra that allows the translation of words into mathematical language. This skill is fundamental for problem-solving across mathematics and sciences. When writing expressions, it's important to correctly interpret the words given in a problem.

• Consider the order of operations. Words such as 'add', 'subtract', 'multiply', and 'divide' need to be accurately translated into mathematical symbols.
• Understand conjunctions like "and" and "from" as they dictate the sequence of operations.

For instance, the phrase "the reciprocal of \(x\), subtracted from the reciprocal of 5" becomes \(\frac{1}{5} - \frac{1}{x}\) in mathematical terms. Mastery of writing expressions is indispensable because it serves as the groundwork for not just understanding problems, but also for devising solutions and exploring deeper algebraic functions.