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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 58
Chapter 6, Problem 58

Exercises 57–59 will help you prepare for the material covered in the next section. Add: (5x−3)/(x2+1) + 2x/(x2+1)2.

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Identify the two rational expressions to be added: 5x−3x2+1 and 2xx2+12.
Determine the least common denominator (LCD) for the two fractions. Since the denominators are x^2+1 and (x^2+1)^2, the LCD is (x^2+1)^2.
Rewrite the first fraction so that it has the LCD as its denominator by multiplying both its numerator and denominator by (x^2+1). This gives: (5x−3)(x^2+1)x2+12.
Now that both fractions have the same denominator, combine the numerators over the common denominator: (5x−3)(x^2+1) + 2xx2+12.
Expand the numerator by distributing and then combine like terms if possible. The expression is now ready for any further simplification or factoring.

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Key Concepts

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

Adding Rational Expressions

Adding rational expressions involves combining fractions with algebraic expressions in the numerator and denominator. When denominators are the same or related, you can add the numerators directly or after adjusting to a common denominator.
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Polynomial Denominators and Factoring

Understanding the structure of polynomial denominators, such as recognizing powers like (x^2 + 1) and (x^2 + 1)^2, is essential. This helps in identifying common denominators and simplifying expressions before addition.
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Simplifying Algebraic Fractions

After adding rational expressions, simplifying the resulting fraction by combining like terms and factoring if possible is important. This ensures the final answer is in its simplest form for clarity and correctness.
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Related Practice
Textbook Question

Exercises 57–59 will help you prepare for the material covered in the next section. Subtract: 3x−4−2x+2\(\frac{3}{x - 4}\) - \(\frac{2}{x + 2}\)

Textbook Question

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

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Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

{x≥0y≥02x+5y<103x+4y≤12\(\begin{cases}\)x \(\geq\) 0 \(\y\) \(\geq\) 0 \\2x + 5y < 10 \\3x + 4y \(\leq\) 12\(\end{cases}\)

Textbook Question

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

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Textbook Question

Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.

Textbook Question

Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.