Problem 57 - Algebra A Combined Function

Question

Write each phrase as an expression. The reciprocal of \(x\), added to the reciprocal of 2

Step-by-Step Solution

Answer

The expression is \(\frac{1}{x} + \frac{1}{2}\).

Step 1: Understand Reciprocals

A reciprocal of a number is 1 divided by that number. In mathematical terms, the reciprocal of a number \(a\) is given by \(\frac{1}{a}\). For instance, the reciprocal of \(x\) is \(\frac{1}{x}\).

Step 2: Identify Each Reciprocal

Identify the reciprocals in the phrase provided: 'the reciprocal of \(x\)' and 'the reciprocal of 2'. These correspond to \(\frac{1}{x}\) and \(\frac{1}{2}\), respectively.

Step 3: Add the Reciprocals

To express the entire phrase as an expression, add the two reciprocals. The expression is written as the sum: \(\frac{1}{x} + \frac{1}{2}\).

Key Concepts

Reciprocal

In mathematics, understanding the concept of a reciprocal is key to solving problems involving division and fractions.
The reciprocal of a number is simply 1 divided by that number. It's like flipping the number upside down when it is a fraction. For example, the reciprocal of 5 is \(rac{1}{5}\). If you have a fraction like \(rac{3}{4}\), its reciprocal would be \(rac{4}{3}\).
Here’s why reciprocals are important:

• When you multiply a number by its reciprocal, the result is always 1. For instance, \(rac{5}{1} imes rac{1}{5} = 1\).
• Reciprocals are used to change division into multiplication, making calculations more manageable.

Addition of Fractions

Adding fractions may initially seem complicated, but it is straightforward once you understand the rules. To add fractions, they must have the same denominator.
If they don’t, you must find a common denominator before adding. For instance, when adding \(rac{1}{x}\) and \(rac{1}{2}\), you need to make sure the denominators are equal.
To do this, find the least common multiple of the denominators:

• The least common multiple (LCM) of \(x\) and \(2\) is \(2x\).
• Convert each fraction to have this common denominator. \(rac{1}{x} = rac{2}{2x}\) and \(rac{1}{2} = rac{x}{2x}\).
• Now, you can add them: \(\frac{2}{2x} + \frac{x}{2x} = \frac{2+x}{2x}\).

By following these steps, you can add any fractions effectively.

Mathematical Phrases

Translating mathematical phrases into expressions is like learning a new language. It involves understanding words that describe mathematical operations or relationships.
For example, the phrase "the reciprocal of \(x\), added to the reciprocal of 2," becomes \(\frac{1}{x} + \frac{1}{2}\). Every word in the phrase corresponds to a component of the mathematical expression.
Here’s a simple breakdown:

• The phrase "reciprocal of" converts a number into its reciprocal. For example, "reciprocal of \(x\)" is \(\frac{1}{x}\).
• "Added to" means you need to add, which is represented by the plus sign "+".
• Identify each part of the phrase with its respective mathematical symbol or operation to form a complete expression.

By practicing the conversion of phrases into expressions, solving math problems becomes a smoother process.