In Exercises 67–70, find all values of x such that y = 0. $y = \frac{x + 6}{3x - 12} - \frac{5}{x - 4} - \frac{2}{3}$

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1. Start with the given equation: \(y = \frac{x + 6}{3x - 12} - \frac{5}{x - 4} - \frac{2}{3}\). We want to find all values of \(x\) such that \(y = 0\), so set the equation equal to zero: \(\frac{x + 6}{3x - 12} - \frac{5}{x - 4} - \frac{2}{3} = 0\).
2. Notice that \$3x - 12\( can be factored as \)3(x - 4)\(. Rewrite the equation using this factorization: \)\frac{x + 6}{3(x - 4)} - \frac{5}{x - 4} - \frac{2}{3} = 0$.
3. To combine the fractions, find the least common denominator (LCD). The denominators are \$3(x - 4)\(, \)x - 4\(, and \)3\(. The LCD is \)3(x - 4)$. Rewrite each term with this common denominator:
4. \(\frac{x + 6}{3(x - 4)} - \frac{5 \cdot 3}{3(x - 4)} - \frac{2(x - 4)}{3(x - 4)} = 0\).
5. Combine the numerators over the common denominator:
6. \(\frac{(x + 6) - 15 - 2(x - 4)}{3(x - 4)} = 0\).
7. Since a fraction equals zero only when its numerator is zero (and the denominator is not zero), set the numerator equal to zero and solve for \(x\): \((x + 6) - 15 - 2(x - 4) = 0\). Then simplify and solve the resulting linear equation.

Key Concepts

Solving Rational Equations

Solving rational equations involves finding values of the variable that satisfy an equation containing fractions with polynomials in the numerator and denominator. The key step is to find a common denominator to combine terms or clear denominators by multiplying both sides, then solve the resulting equation.

Domain Restrictions and Excluded Values

When working with rational expressions, certain values of the variable make denominators zero, which are undefined. Identifying these excluded values is essential to avoid invalid solutions and to restrict the domain of the function before solving the equation.

Setting the Function Equal to Zero

To find values of x such that y = 0, set the entire expression equal to zero and solve for x. This means finding when the numerator of the simplified rational expression equals zero, while ensuring the denominator is not zero, to determine the roots of the function.