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Chapter 7: Problem 68

True or False? \(10 \div x=\frac{1}{10} \cdot x\). Justify your answer.

Short Answer

Expert verified
False, \(10 \div x\) is not equal to \(\frac{1}{10} \cdot x\) for all values of \(x\). This only holds true when \(x= \sqrt{10}\).

Step by step solution

01

Analyze and Simplify the given equation

Let's start by simplifying the equation \(10 \div x=\frac{1}{10} \cdot x\). As we know, division by a number is the same as multiplication by its reciprocal, so we can rewrite \(10 \div x\) as \(10 \cdot \frac{1}{x}\). Now the equation looks like this: \(10 \cdot \frac{1}{x}=\frac{1}{10} \cdot x\).
02

Convert to common form

For better comparison, convert both parts of the equation to the form of 'number times x'. The left hand side (LHS) is already in this form, so we focus on the right hand side (RHS). Multiply \(\frac{1}{10}\) with \(x\) to get \( \frac{1}{10}x\).
03

Compare both sides of the equation

Now, both sides of the equation are in a 'number times x' format, i.e., \(10 \cdot \frac{1}{x}\) and \(\frac{1}{10}x\). As we can see, it is not possible for \(10 \cdot \frac{1}{x}\) to be equal to \(\frac{1}{10}x\) for any value of \(x\) except \(x= \sqrt{10}\). However, this is not generally true for all \(x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocals
In algebra, reciprocals often play a key role when dealing with division and multiplication. The reciprocal of a number is simply 1 divided by that number.
For example, the reciprocal of 10 is \( \frac{1}{10} \). When you multiply a number by its reciprocal, you always get 1. Let's see how this concept applies to dividing: instead of dividing by a number, you can multiply by its reciprocal.
  • Division by \( x \) is the same as multiplying by \( \frac{1}{x} \).
  • This transformation often simplifies complex expressions.
Understanding reciprocals helps us transform equations to more manageable forms, creating a clearer path toward finding solutions.
Equation Simplification
Simplifying equations is an essential skill in algebra. It involves rewriting equations in a more straightforward, yet mathematically equivalent, form, often by reducing fractions or rationalizing terms.
In the provided exercise, simplifying the equation \( 10 \div x = \frac{1}{10} \cdot x \) was achieved by transforming the expression involving division into one involving multiplication and reciprocals.
  • The left-hand side \( 10 \div x \) was rewritten as \( 10 \cdot \frac{1}{x} \), utilizing the concept of reciprocals.
  • Both sides were simplified to a 'number times \( x \)' format for easy comparison.
Simplifying makes equations easier to solve or analyze, by reducing them to simpler components that are easy to compare or manipulate.
Comparing Algebraic Expressions
When comparing algebraic expressions, it's crucial to ensure they are in the same form to accurately determine their equality. This often involves algebraic manipulation such as simplifying, factoring, or expanding expressions.
In the equation \( 10 \cdot \frac{1}{x} = \frac{1}{10}x \), simplifying both sides to a 'number times \( x \)' form was necessary for an accurate comparison.
  • The left expression \( 10 \cdot \frac{1}{x} \) and the right expression \( \frac{1}{10}x \) aren't equal under normal circumstances.
  • The expressions share equality only under specific conditions (e.g., \( x = \sqrt{10} \)), which doesn鈥檛 hold true universally for all \( x \).
By carefully examining the structure of each side, we can better understand when and how two expressions can be equivalent.

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