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Chapter 7: Problem 53

Perform the indicated operation. Simplify, if possible. $$ \frac{4 x}{5}+\frac{x+3}{-5} $$

Short Answer

Expert verified
\[\frac{3(x - 1)}{5}\]

Step by step solution

01

Identify the common denominator

Both fractions have denominators involving 5. Rewrite the equation with a common denominator: \[\frac{4x}{5} + \frac{x + 3}{-5}\]
02

Simplify the second fraction

Factor and simplify the second fraction to align it with the same sign denominator: \[\frac{4x}{5} + \frac{-x - 3}{5}\]
03

Combine the fractions

Now that both fractions have a common denominator, combine the numerators:\[\frac{4x - (x + 3)}{5}\]
04

Simplify the numerator

Distribute the negative sign and combine like terms:\[\frac{4x - x - 3}{5} = \frac{3x - 3}{5}\]
05

Factor the numerator

Factor out the common factor in the numerator:\[\frac{3(x - 1)}{5}\]

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Key Concepts

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

Common Denominator
When dealing with fraction operations, finding a common denominator is a critical first step. The common denominator is a multiple shared by all denominators in the fractions you are working with. This allows you to rewrite each fraction so that they all have the same denominator, making it simpler to add, subtract, or compare the fractions. In our exercise, both fractions already have denominators involving 5. So, the common denominator is simply 5. This step ensures consistency and forms the basis for combining fractions correctly.
Simplifying Fractions
Simplifying fractions involves rewriting them in their simplest form. This can mean factoring out the greatest common divisor from the numerator and the denominator, or merely adjusting the signs in the fractions to match. In our exercise, we simplify the second fraction by adjusting its sign to align with the common denominator: \( \frac{x+3}{-5} = \frac{-x-3}{5} \). This step essentially helps in reducing the complexity of the problem, making the later steps more manageable.
Combining Fractions
Combining fractions is the process of adding or subtracting fractions, as needed, once they share a common denominator. After ensuring the fractions have the same denominator, combine the numerators directly over this common denominator. In our example, the combined fraction looks like this: \( \frac{4x}{5} + \frac{-x-3}{5} = \frac{4x - (x + 3)}{5} \). It's essential to handle the numerators carefully, especially when managing signs, to avoid mistakes.
Factoring
Factoring is an algebraic process where you express a polynomial as a product of its factors. This step is crucial in simplifying expressions further. After combining the fractions in our exercise, the numerator \(3x - 3\) can be factored. This involves identifying common terms and factoring them out, resulting in: \( \frac{3(x-1)}{5} \). Factoring helps to simplify the equation further, making it easier to understand and solve.

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