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Chapter 1: Problem 74

Write each expression in two equivalent forms, as in Example \(6 .\) $$ \frac{9}{-14} $$

Short Answer

Expert verified
\frac{9}{-14} = \frac{-9}{14} = -\frac{9}{14}

Step by step solution

01

Understand the Problem

We need to write the fraction \(\frac{9}{-14}\) in two equivalent forms. Equivalent forms should represent the same value as the original fraction.
02

Equivalent Form Using Negative Sign in the Numerator

One equivalent form can be obtained by moving the negative sign from the denominator to the numerator: \(\frac{9}{-14} = \frac{-9}{14}\).
03

Equivalent Form Using Negative Sign in Front of the Fraction

Another equivalent form can be obtained by placing the negative sign in front of the entire fraction: \(\frac{9}{-14} = -\frac{9}{14}\).

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Key Concepts

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

Negative Fractions
Negative fractions can sometimes be tricky, but they follow the same rules as regular fractions. When dealing with negative fractions, it's important to remember that the position of the negative sign doesn't change the value of the fraction.

For example, \(\frac{9}{-14}\) can be rewritten in two equivalent forms:
  • By moving the negative sign to the numerator: \(\frac{-9}{14}\).

  • By placing the negative sign in front of the entire fraction: \(-\frac{9}{14}\).
Both expressions represent the same value: a fraction that is less than zero.
Understanding this concept can help to simplify and work with negative fractions more easily in various mathematical problems and solutions.
Fraction Representation
Fractions represent parts of a whole. The way they are written can vary, but they always show the same value if they are equivalent.

The fraction \(\frac{9}{-14}\) has two parts:
  • The numerator (top part): 9

  • The denominator (bottom part): -14
Changing the position of the negative sign doesn't affect the value. Whether the negative sign is in the numerator, denominator, or in front of the fraction, the fraction still represents the same quantity.

For example, consider the three forms of the fraction \(\frac{9}{-14}\):
  • \(\frac{9}{-14}\)

  • \(\frac{-9}{14}\)

  • \(-\frac{9}{14}\)
Each form is a different way to represent the same fraction.
Learning how to correctly represent fractions can help students solve problems more efficiently.
Numerator and Denominator
Understanding the terms numerator and denominator is crucial for working with fractions.

The numerator represents the number of parts we have, while the denominator shows how many parts make up a whole.
For example, in the fraction \(\frac{9}{-14}\):
  • The numerator is 9, which tells us there are 9 parts.

  • The denominator is -14, indicating that the total number of parts in a whole is represented as -14 due to the negative sign.
If we move the negative sign to different parts of the fraction, the concept remains the same:
  • In \(\frac{-9}{14}\), the numerator is -9 and the denominator is 14.

  • In \(-\frac{9}{14}\), the negative sign applies to the entire fraction.
These all show that the fraction represents a total value less than zero.
By mastering the concepts of numerator and denominator, students can better understand how fractions work.

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Most popular questions from this chapter

Simplify. $$ |10(-5)|+1(-1) $$

Simplify. $$ \frac{5^{2}-3^{2}}{2 \cdot 6-4} $$

SYNTHESIS Write the sentence \((-x)^{2} \neq-x^{2}\) in words. Explain why \((-x)^{2}\) and \(-x^{2}\) are not equivalent.

Evaluate using a calculator. $$ 3(m+2 n) \div m, \text { for } m=1.6 \text { and } n=5.9 $$

Solve. Consider only integer replacements. $$|x|<3$$
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