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Chapter 8: Problem 69

Solve. [ 7.6] $$ \frac{x}{(x-3)(x+7)}=0 $$

Short Answer

Expert verified
x = 0

Step by step solution

01

Identify the equation

We start with the given equation: \[\frac{x}{(x-3)(x+7)}=0\]
02

Understand the zero property

In fractions, for the entire fraction to be zero, the numerator must be zero. The denominator does not affect whether the fraction is zero. Therefore, we focus on the numerator.
03

Set the numerator to zero

Set the numerator equal to zero: \[x = 0\]
04

Verify the solution

Check that substituting \(x = 0\) does not make the denominator zero; substituting \(x = 0\) gives: \[(x-3)(x+7) = (-3)(7) = -21 \]Since the denominator is not zero, \(x = 0\) is a valid solution.

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Key Concepts

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

The Zero Property
The Zero Property is a fundamental concept in algebra. It tells us that for a fraction to be zero, its numerator must be zero. The denominator, while important for other operations, does not influence whether the fraction equals zero. In the context of our exercise, \(\frac{x}{(x-3)(x+7)}=0\), we need to focus on the numerator, which is the part of the fraction above the dividing line. Here, our numerator is just \(x\). To make the entire fraction zero, we set the numerator equal to zero. This means we solve for \(x \) by setting \(\frac{x}{(x-3)(x+7)}=0\) to \(x=0\).
Understanding the Numerator
The term 'numerator' refers to the top part of a fraction. It indicates how many parts of the whole are being considered. In our example equation \(\frac{x}{(x-3)(x+7)}=0\), \(x\) is the numerator. When solving the fraction for zero, the zero property tells us that only the numerator needs to be zero. To do this, we simply set \(x = 0\). Let's verify the solution by substituting \(x = 0\) back into the equation. Doing so, we get: \(0 \), which indeed confirms that the numerator is zero, validating our solution.
The Role of the Denominator
The 'denominator' is the bottom part of a fraction. In our equation, \((x-3)(x+7)\) is the denominator. The denominator helps us understand how many equal parts the whole is divided into, but for finding when the entire fraction equals zero, it doesn't affect the process directly. However, it is still crucial to check that the solution for the numerator does not make the denominator zero, since division by zero is undefined. For \(x = 0\): \((x-3)(x+7) = (-3)(7) = -21\), which is non-zero. Therefore, \(x=0\) is not only a solution due to the numerator being zero, but it also keeps the denominator valid.

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