Ch. R - Algebra Review

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. R - Algebra Review

Problem 91

Chapter 1, Problem 91

Simplify each complex fraction. See Examples 5 and 6. [(y + 3)/y − 4/(y − 1)] / [(y/y + 1/y) / (y − 1) + 1/y]

Verified step by step guidance

Rewrite the complex fraction clearly by expressing the numerator and denominator separately. The numerator is \(\frac{y + 3}{y} - \frac{4}{y - 1}\) and the denominator is \(\frac{y}{y - 1} + \frac{1}{y}\).

Find a common denominator for the terms in the numerator. The denominators are \(y\) and \(y - 1\), so the common denominator is \(y(y - 1)\). Rewrite each fraction with this common denominator.

Combine the fractions in the numerator by subtracting the numerators over the common denominator \(y(y - 1)\).

Similarly, find a common denominator for the terms in the denominator, which are \(y - 1\) and \(y\). The common denominator is also \(y(y - 1)\). Rewrite and add the fractions in the denominator.

Now, rewrite the entire complex fraction as a division of two fractions with the same denominator \(y(y - 1)\). When dividing, multiply the numerator fraction by the reciprocal of the denominator fraction, then simplify the resulting expression by canceling common factors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by finding common denominators or multiplying numerator and denominator by the least common denominator (LCD) to eliminate the smaller fractions.

Finding the Least Common Denominator (LCD)

The LCD is the smallest expression that all denominators in a problem divide into evenly. Identifying the LCD allows you to combine or simplify fractions by converting them to equivalent fractions with a common denominator, which is essential for simplifying complex fractions.

Algebraic Manipulation of Rational Expressions

This involves applying algebraic techniques such as factoring, expanding, and canceling common factors in expressions with variables in numerators and denominators. Mastery of these skills is crucial for simplifying complex fractions that contain polynomial expressions.

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Related Practice

Textbook Question

Add or subtract, as indicated. See Example 6. √6 + √6

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Textbook Question

Add or subtract, as indicated. See Example 6. 5√3 + √12

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Textbook Question

Simplify each inequality if needed. Then determine whether the statement is true or false. -6 < 7 + 3

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Textbook Question

(Modeling) Distance from the Origin of the Nile River The Nile River in Africa is about 4000 mi long. The Nile begins as an outlet of Lake Victoria at an altitude of 7000 ft above sea level and empties into the Mediterranean Sea at sea level (0 ft). The distance from its origin in thousands of miles is related to its height above sea level in thousands of feet (x) by the following formula.

Distance = (7 − x) / (0.639x + 1.75)

For example, when the river is at an altitude of 600 ft, x = 0.6 (thousand), and the distance from the origin is

Distance ≈ (7 − 0.6) / (0.639 × 0.6 + 1.75) ≈ 3, which represents 3000 mi. (Data from The World Almanac and Book of Facts.) What is the distance from the origin of the Nile when the river has an altitude of 7000 ft?

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Textbook Question

Add or subtract, as indicated. See Example 6. √6 + √7

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Textbook Question

Simplify each inequality if needed. Then determine whether the statement is true or false. 7 ≤ 7

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