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Chapter 7: Problem 52

Add or subtract. Write answer in lowest terms. \(\frac{10}{m-2}+\frac{5}{2-m}\)

Short Answer

Expert verified
\(\frac{5}{m-2}\)

Step by step solution

01

- Understand the Denominators

Recognize that the denominators \(m-2\) and \(2-m\) are almost the same except for a negative sign. They can be rewritten to have a common form.
02

- Rewrite the Second Fraction

Rewrite \(\frac{5}{2-m}\). Since \(2-m = -(m-2)\), \(\frac{5}{2-m} = \frac{5}{-(m-2)} = -\frac{5}{m-2}\).
03

- Combine the Fractions

Now that both fractions have the same denominator, combine them: \[\frac{10}{m-2} + (-\frac{5}{m-2}) = \frac{10 - 5}{m-2} = \frac{5}{m-2}\].
04

- Simplify the Expression

The fraction \(\frac{5}{m-2}\) is already in its lowest terms. No further simplification is required.

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Key Concepts

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

common denominators
When adding or subtracting fractions, having the same denominator is essential. Denominators are the bottom parts of fractions, and ensuring they match lets us easily combine the fractions.

In this exercise, the denominators are \(m-2\) and \(2-m\). Notice that if we multiply \(2-m\) by -1, it becomes \(-(m-2)\). This reveals that they are essentially the same denominator up to a sign change.

To make the denominators the same, we can rewrite fractions. For example, \( \frac{5}{2-m} \) can be transformed into \( \frac{-5}{m-2} \) by multiplying both the numerator and the denominator by -1. Now, we have a common denominator: \(m-2\).
simplifying fractions
Simplifying fractions means making them easier to understand or reducing them to their smallest form. This involves combining terms and reducing both the numerator and the denominator by any common factors they may have.

For example, \(10 - 5 = 5\). When combined over the common denominator, \(m-2\), it becomes \ \frac{5}{m-2}\.

This fraction is already in its lowest terms because there are no common factors between the numerator (5) and the denominator (m-2).

Simplification helps in easily reading and interpreting fractions, making math problems simpler to solve.
negative signs in fractions
Negative signs can sometimes be tricky, but they follow simple rules. They can appear in either the numerator or the denominator but are generally moved to the front of the fraction for clarity.

In our original problem, \( \frac{5}{2-m} \) could be confusing. We recognize that \2-m = -(m-2)\. Therefore, \( \frac{5}{2-m} = \frac{5}{-(m-2)} = -\frac{5}{m-2}\). You move the negative sign to the numerator to make it easier to handle.

Proper handling of negative signs ensures that your fraction operations are accurate. It maintains the integrity and value of your calculations.

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