Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 10

Chapter 6, Problem 10

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

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Identify the form of the rational expression. Here, the denominator is a product of two linear factors:

x (x - 1)

Set up the partial fraction decomposition by expressing the rational expression as a sum of fractions with unknown constants in the numerators over each linear factor:

\(\frac{1}{x(x-1)}\) = \(\frac{A}{x}\) + \(\frac{B}{x-1}\)

, where

A

and

B

are constants to be determined.

Multiply both sides of the equation by the common denominator

x(x-1)

to clear the denominators:

1 = A(x-1) + Bx

Expand the right side:

1 = A x - A + B x

, then combine like terms:

1 = (A + B) x - A

Equate the coefficients of corresponding powers of

x

on both sides to form a system of equations: For the

x

term,

0 = A + B

; for the constant term,

1 = -A

. These equations can be solved to find

A

and

B

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Key Concepts

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

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with simpler denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Polynomials

Factoring polynomials means rewriting a polynomial as a product of its factors. Recognizing and factoring denominators into linear or irreducible quadratic factors is crucial for setting up the correct form of partial fractions.

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Related Practice

Textbook Question

Solve each system in Exercises 5–18. $\begin{cases}\)3x + 2y - 3z = -2 \\2x - 5y + 2z = -2 \\4x - 3y + 4z = 10\(\end{cases}$

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\)xy = 6 \\2x - y = 1\(\end{cases}$

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Textbook Question

Write the partial fraction decomposition of each rational expression. (3x +50)/(x -9)(x +2)

Textbook Question

The perimeter of a table tennis top is 28 feet. The difference between 4 times the length and 3 times the width is 21 feet. Find the dimensions.

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Textbook Question

Graph each inequality. x≤−3

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Textbook Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

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