Questions: 21945, Fall 2024 Rational Expressions and Dividing Question 19, 6.1.141 Determine whether the following statement is true or false. If the statement is false, make the necessary change. The domain of f(x) = 5/(x(x-6)+8(x-6)) is (-∞, 6) ∪ (6, ∞). Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The statement is false. The domain of f(x) is (Type your answer in interval notation.) B. The statement is true.

Solution Steps

1. Simplify the given rational expression \( f(x) = \frac{5}{x(x-6) + 8(x-6)} \).
2. Combine like terms in the denominator.
3. Identify the values of \( x \) that make the denominator zero, as these values are excluded from the domain.
4. Determine the domain of \( f(x) \) by excluding these values from the set of all real numbers.

We start with the function \( f(x) = \frac{5}{x(x-6) + 8(x-6)} \). First, we simplify the denominator:

\[ x(x-6) + 8(x-6) = x^2 - 6x + 8x - 48 = x^2 + 2x - 48 \]

Next, we need to find the values of \( x \) that make the denominator zero. We solve the equation:

\[ x^2 + 2x - 48 = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2, c = -48 \):

\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-48)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 192}}{2} = \frac{-2 \pm \sqrt{196}}{2} = \frac{-2 \pm 14}{2} \]

This gives us the solutions:

\[ x = \frac{12}{2} = 6 \quad \text{and} \quad x = \frac{-16}{2} = -8 \]

The values \( x = 6 \) and \( x = -8 \) make the denominator zero, so these values are excluded from the domain. Therefore, the domain of \( f(x) \) is:

\[ (-\infty, -8) \cup (-8, 6) \cup (6, \infty) \]

Final Answer