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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 89a
Chapter 2, Problem 89a

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x/(x + 3) - 12/(x - 3) = (4x2 + 36)/(x2 - 9)

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Rewrite the equation to identify the denominators. Notice that the denominators are \(x + 3\), \(x - 3\), and \(x^2 - 9\). Recognize that \(x^2 - 9\) is a difference of squares and can be factored as \((x + 3)(x - 3)\).
Multiply through the entire equation by the least common denominator (LCD), which is \((x + 3)(x - 3)\), to eliminate the fractions. This will simplify the equation significantly.
Distribute the LCD to each term in the equation. For the first term, \(\frac{4x}{x + 3}\), multiplying by \((x + 3)(x - 3)\) leaves \(4x(x - 3)\). For the second term, \(\frac{-12}{x - 3}\), multiplying by \((x + 3)(x - 3)\) leaves \(-12(x + 3)\). For the right-hand side, \(\frac{4x^2 + 36}{x^2 - 9}\), multiplying by \((x + 3)(x - 3)\) leaves \(4x^2 + 36\).
Simplify the resulting equation by distributing and combining like terms. Expand \(4x(x - 3)\) and \(-12(x + 3)\), then combine all terms on one side of the equation to form a standard quadratic equation.
Solve the quadratic equation using factoring, the quadratic formula, or completing the square. After finding the solutions, check for any restrictions on \(x\) (e.g., \(x \neq -3\) and \(x \neq 3\) because these values make the original denominators undefined). Determine whether the equation is an identity (true for all values of \(x\)), a conditional equation (true for specific \(x\) values), or inconsistent (no solution).

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Key Concepts

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

Types of Equations

In algebra, equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity holds true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solutions. Understanding these classifications is crucial for determining the nature of the given equation.
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Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. In the given equation, the presence of rational expressions requires careful manipulation, such as finding a common denominator or simplifying the expressions. Mastery of operations with rational expressions is essential for solving the equation accurately.
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Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, yield the original polynomial. In this problem, recognizing that the denominator x^2 - 9 can be factored into (x + 3)(x - 3) is vital for simplifying the equation and solving it effectively.
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Related Practice
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Textbook Question

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4/(x2 + 3x - 10) - 1/(x2 + x - 6) = 3/(x2 - x - 12)

Textbook Question

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