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Chapter 7: Problem 39

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+1}-\frac{1}{x}}{\frac{1}{x+1}} $$

Short Answer

Expert verified
-\frac{1}{x}

Step by step solution

01

- Simplify the numerator

The numerator is \(\frac{1}{x+1} - \frac{1}{x}\). To combine the fractions, find a common denominator: \((x+1)x\). Rewrite both fractions with the common denominator: \[ \frac{1 \times x}{(x+1)x} - \frac{1 \times (x+1)}{(x+1)x} \] This results in: \[ \frac{x - (x+1)}{(x+1)x} \]
02

- Simplify the expression in the numerator

Simplify the numerator: \[ \frac{x - x - 1}{(x+1)x} = \frac{-1}{(x+1)x} \]
03

鈥� Divide the numerator by the denominator

The expression we have now is \(\frac{\frac{-1}{(x+1)x}}{\frac{1}{x+1}}\). When dividing by a fraction, multiply by its reciprocal:\[ \frac{-1}{(x+1)x} \times \frac{x+1}{1} = \frac{-1 \times (x+1)}{(x+1)x} \]
04

鈥� Simplify the final expression

Cancel out the common \((x+1)\) in the numerator and denominator:\[ \frac{-1}{x} \]

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Key Concepts

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

common denominator
When working with complex fractions, having a common denominator is crucial in simplifying terms. In the provided exercise, the numerator is \(\frac{1}{x+1} - \frac{1}{x}\). To combine these fractions, find a common denominator. For \(\frac{1}{x+1}\) and \(\frac{1}{x}\), the common denominator is \( (x+1)x \). This is because it ensures both denominators can be expressed equivalently with the least common multiple. Rewrite the fractions with the common denominator to facilitate their combination. \(\frac{1 \times x}{(x+1)x} - \frac{1 \times (x+1)}{(x+1)x} = \frac{x - (x+1)}{(x+1)x}\). This step is essential for simplification, as it prepares the fractions for subtraction.
fraction division
Division of fractions follows a straightforward rule: multiply by the reciprocal. In our exercise, the expression transitions to \( \frac{\frac{-1}{(x+1)x}}{\frac{1}{x+1}} \). This division is simplified by multiplying by the reciprocal of the divisor fraction. Hence, it reformed to \(\frac{-1}{(x+1)x} \times \frac{x+1}{1} \). This step effectively turns the division problem into a multiplication one, a simpler and more manageable format. Remember this rule: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
canceling common factors
Cancelling common factors greatly simplifies fractions. Once multiplication in the previous step produces \(\frac{-1 \times (x+1)}{(x+1)x} \), observe the \( (x+1) \) in both the numerator and the denominator. These are common factors and can be cancelled out. Simplifying the expression results in \(\frac{-1}{x} \). Cancelling out these common factors streamlines the fraction, preserving the mathematical equivalence but in a more straightforward form. Always look for such factors to simplify your expressions.
elementary algebra
Elementary algebra forms the basis for understanding the operations carried out. It encompasses a wide range of concepts and operations, including manipulation of fractions. In the given exercise, knowledge of how to combine fractions using a common denominator, how to rewrite terms, and how to simplify by cancelling factors, all fall under elementary algebra. Having a strong grasp of these concepts allows for smoother and more intuitive problem-solving processes. Remember, breaking problems into smaller, manageable steps often reveals the path to the solution more clearly.

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Most popular questions from this chapter

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=5\).

For exercises 43-58, (a) solve. (b) check. $$ \frac{2}{x}=0 $$

For exercises 79-82, (a) clear the fractions and solve. (b) check. $$ 1=\frac{7}{6} w+\frac{5}{12} $$

The relationship of the amount of salad dressing, \(x\), and the amount of sodium in the dressing, \(y\), is a direct variation. Six servings of dressing contain \(1800 \mathrm{mg}\) of sodium. a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the amount of sodium in a bottle that contains 16 servings of salad dressing. d. Use slope-intercept graphing to graph this equation. e. Use the graph to find the amount of sodium in 3 servings of salad dressing.

For exercises 43-58, (a) solve. (b) check. $$ \frac{4}{a+6}=\frac{9}{a-4} $$
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