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Chapter 7: Problem 59

\(\frac{8-3 x}{x+2}+\frac{4+2 x}{x+2}\)

Short Answer

Expert verified
\(\frac{12 - x}{x + 2}\)

Step by step solution

01

Equivalent Fraction

First, recognize that both fractions have identical denominators. This allows the fractions to be combined into a single fraction with the denominator \(x + 2\). The combined fraction is then \(\frac{8 - 3x + 4 + 2x}{x + 2}\).
02

Simplify the Numerator

Next, simplify the numerator of the fraction by combining like terms. The \(8\) and \(4\) are constants and can be added together, and the \(-3x\) and \(2x\) are like terms and can be added together. This simplifies to \(\frac{12 - x}{x + 2}\).
03

Final Form

Lastly, recognize if the numerator and denominator have any common factors that can be divided out. In this case, they do not. So the final simplified expression is \(\frac{12 - x}{x + 2}\).

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Key Concepts

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

Equivalent Fractions
Recognizing equivalent fractions is very handy in algebra, especially when dealing with expressions. When fractions have the same denominator, we can easily add or subtract them. In the given exercise, both fractions have the denominator of \(x+2\).

This means they are already set up to be combined. Combine the numerators, treating each part as a piece of the larger puzzle, while keeping the denominator the same:
  • Add up all terms in the numerators: \(8 - 3x\) plus \(4 + 2x\).
  • Keep the shared denominator \(x+2\) unchanged.
The expression becomes a single fraction: \[\frac{8 - 3x + 4 + 2x}{x + 2}\] This process simplifies multiple fractions into one, making future steps a neat affair.
Simplifying Expressions
Simplifying algebraic expressions is all about making the expression more understandable by reducing complexity. Once we have a single fraction, we should focus on tidying up the numerator. Don't be intimidated by all the different components. Take it step by step.

Focus on similar items to make things simpler:
  • Combine the constant numbers together. Here, \(8\) and \(4\) combine to make \(12\).
  • Similarly, combine the \(-3x\) and \(2x\) terms. They simplify to \(-x\).
Thus, we end up with a much cleaner expression:\[\frac{12 - x}{x + 2}\]The process of simplifying keeps the essence of the expression but trims down unnecessary complications.
Combining Like Terms
Combining like terms is an essential skill in algebra that helps simplify and solve equations more effectively. "Like terms" are those which have the same variable raised to the same power. In our case:
  • The numbers without variables are like terms. Hence, \(8\) and \(4\) can be added together.
  • Terms with the variable \(x\), such as \(-3x\) and \(2x\), are also alike because they share the variable part.
Add up the constants and then combine the variable terms:- \(8 + 4 = 12\)- \(-3x + 2x = -x\)
As a result, the expression becomes clearer and more streamlined, looking like this:\[\frac{12 - x}{x + 2}\] Using these steps, even complex expressions become manageable with practice and patience.

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Most popular questions from this chapter

\(\frac{\left(\frac{x^{5}}{12}\right)}{\left(\frac{x^{2}}{54}\right)}\)

\(\frac{\left(\frac{4}{9}\right)}{\left(\frac{2}{39}\right)}\)

\(\frac{5}{2 x}\) and \(\frac{x+1}{5}\)

Electronics When three resistors of resistance \(R_{1}, R_{2}\), and \(R_{3}\) (all in ohms) are connected in parallel, the total resistance (in ohms) is modeled by $$ \frac{1}{\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\right)} $$ Simplify this complex fraction.

\(\frac{3 x}{x+1}=\frac{2}{x-1}\)
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