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Chapter 7: Problem 63

Perform each indicated operation. $$ \frac{6}{5}+\left(\frac{1}{5}-\frac{8}{5}\right) $$

Short Answer

Expert verified
\( \frac{-1}{5} \)

Step by step solution

01

Solve Inside the Parentheses

Firstly, let's focus on solving the operation inside the parentheses: \( \frac{1}{5} - \frac{8}{5} \). Since both fractions have the same denominator, we can subtract the numerators directly: \( 1 - 8 = -7 \). Thus, \( \frac{1}{5} - \frac{8}{5} = \frac{-7}{5} \).
02

Add the Result to the Outside Fraction

Now we add \( \frac{6}{5} \) to the result obtained in Step 1, which is \( \frac{-7}{5} \). Again, since both fractions have the same denominator, we add the numerators: \( 6 + (-7) = -1 \). Thus, \( \frac{6}{5} + \frac{-7}{5} = \frac{-1}{5} \).
03

Simplify the Result

Since \( \frac{-1}{5} \) is already in its simplest form, we do not need to simplify further.

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Key Concepts

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

Subtraction of Fractions
When you subtract fractions, the key thing to remember is that you need like denominators. This means both fractions must have the same bottom number. Once they do, it's straightforward: simply subtract the numerators and keep the denominator the same.
For example, in the problem \( \frac{1}{5} - \frac{8}{5} \), you can subtract directly since both fractions have a denominator of 5. Take the numerators (1 and 8) and subtract: \( 1 - 8 = -7 \). The answer becomes \( \frac{-7}{5} \).
  • Ensure denominators match before subtracting.
  • Subtract numerators directly and keep the denominator unchanged.
  • A negative result in the numerator means the result is a negative fraction.
Addition of Fractions
Adding fractions with the same denominator is similar to subtraction. You need like denominators, then you can just add the numerators while keeping the denominator unchanged.
In the exercise, after finding \( \frac{-7}{5} \) from the subtraction, you add \( \frac{6}{5} \) to it. Since both fractions have a denominator of 5, proceed by adding the numerators (6 and -7): \( 6 + (-7) = -1 \). The result is \( \frac{-1}{5} \).
  • The denominator must be the same before adding fractions.
  • The sum of numerators, with the common denominator, gives the result.
  • The answer \( \frac{-1}{5} \) remains negative because the sum of numerators is negative.
Like Denominators
'Like denominators' is a term used when the denominators (the bottom numbers of fractions) are the same. This is crucial for adding or subtracting fractions.
In the example \( \frac{6}{5} + \left( \frac{1}{5} - \frac{8}{5} \right) \), each fraction involved shares a common denominator of 5. Having like denominators lets you perform operations on the numerators directly.
  • Check for like denominators before any addition or subtraction.
  • Having like denominators simplifies the operation to working with just the numerators.
  • If denominators differ, find a common denominator before proceeding.
Without like denominators, the process involves extra steps of finding and converting fractions to their equivalent forms with a common denominator, making your calculations more complex.

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Most popular questions from this chapter

Simplify each complex fraction. See Examples 1 and 2. $$ \frac{\frac{10}{3 x}}{\frac{5}{6 x}} $$

Simplify. $$ \frac{1}{1+(1+x)^{-1}} $$

In your own words, explain how to find the domain of a rational function.

How does the graph of \(y=\frac{x^{2}-9}{x-3}\) compare to the graph of \(y=x+3 ?\) Recall that \(\frac{x^{2}-9}{x-3}=\frac{(x+3)(x-3)}{x-3}=x+3\) as long as \(x\) is not \(3 .\) This means that the graph of \(y=\frac{x^{2}-9}{x-3}\) is the same as the graph of \(y=x+3\) with \(x \neq 3 .\) To graph \(y=\frac{x^{2}-9}{x-3},\) then, graph the linear equation \(y=x+3\) and place an open dot on the graph at \(3 .\) This open dot or interruption of the line at 3 means \(x \neq 3\). (GRAPH CANNOT COPY). $$ \text { Graph } y=\frac{x^{2}-6 x+8}{x-2} $$

\- Solve the following. See Examples I through 7. (Note: Some exercises can be modeled by equations without rational expressions.) A car travels 280 miles in the same time that a motorcycle travels 240 miles. If the car's speed is 10 miles per hour more than the motorcycle's, find the speed of the car and the speed of the motorcycle.
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