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Chapter 1: Problem 773

Simplify. $$ 0 \div\left(x-\frac{1}{2}\right), \text { where } x \neq \frac{1}{2} $$

Short Answer

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Step by step solution

01

Identify the expression

The given expression to simplify is \( 0 \div ( x - \frac{1}{2} ) \).
02

Recall division properties

Any number divided by a non-zero number is zero. Specifically, \( 0 \div a = 0 \) for any \( a eq 0 \).
03

Apply division property to the expression

In our case, \( a = x - \frac{1}{2} \) is not equal to zero, because \( x eq \frac{1}{2} \). So, \( 0 \div ( x - \frac{1}{2} ) = 0 \).

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Key Concepts

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

simplifying expressions
Simplifying an expression means reducing it to its simplest form.
This often makes calculations easier and reveals the core elements of the expression.
Let's break down the steps to simplify a given expression:

  • Identify all parts of the expression. In our case, the expression is given as: \( 0 \div \left( x - \frac{1}{2} \right) \).
  • Apply mathematical properties to simplify. Understanding division properties is crucial here.
  • Recall known rules and apply them. For example, knowing that any number divided by a non-zero number is still 0 helps simplify the given expression to \( 0 \).

Simplifying expressions often involves recognizing patterns and applying mathematical rules to make the expressions more manageable.
division by non-zero
Division by non-zero numbers plays a crucial role in algebra.
It ensures that we do not encounter undefined expressions.
Here鈥檚 what you need to remember:

  • Any non-zero number divided by another non-zero number results in a non-zero number. For example, \(\frac{5}{2} = 2.5\).
  • Zero divided by any non-zero number is always zero: \(0 \div a = 0\) for any \(a 鈮� 0\).
  • Division by zero, however, is undefined and must be avoided. This is why \( x eq \frac{1}{2} \) is crucial in our given exercise.

Applying these principles to our expression, we see that \( x - \frac{1}{2} \eq 0\), simplifying our division to \( 0 \div ( x - \frac{1}{2} ) = 0 \).
algebraic expressions
Algebraic expressions involve variables and constants combined through arithmetic operations.
Here鈥檚 why they matter and how to work with them:

  • Definition: An algebraic expression is a combination of numbers, variables, and arithmetic operations. Example: \( 2x + 3 \).
  • Simplification: To simplify algebraic expressions, combine like terms and perform operations following the rules of algebra.
  • Evaluation: Substitute the values of variables to find the numerical value of the expression. Note: Pay attention to restrictions like \( x 鈮� \frac{1}{2} \) in our exercise.

By understanding and manipulating algebraic expressions, solving a variety of mathematical problems becomes manageable.
In our exercise, recognizing the expression \( x - \frac{1}{2} \) as part of the division allowed us to simplify it correctly, understanding that it cannot be zero ensures a valid and defined result.

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