Question

select the correct answer.

for which quotient is \\(x = 7\\) an excluded value?

\\(\bigcirc \frac{x^2 - 49}{3x + 21} \div \frac{x^2 + 7x}{3x}\\)

\\(\bigcirc \frac{7x}{x^2 - 10x + 21} \div \frac{x + 7}{7}\\)

\\(\bigcirc \frac{x + 7}{x^2 + 6x - 7} \div \frac{7}{2x + 14}\\)

\\(\bigcirc \frac{x - 7}{x^2 + 4x - 21} \div \frac{x^2 + 49}{x + 7}\\)

Explanation:

Identify conditions for excluded values in division of rational expressions

For a quotient of rational expressions \(\frac{A}{B} \div \frac{C}{D}\), the excluded values are those that make any denominator in the original expression or the divisor's numerator equal to zero:

$$ B = 0, \quad D = 0, \quad C = 0 $$

Evaluate each option for \(x = 7\)

• Option 1: \(\frac{x^2 - 49}{3x + 21} \div \frac{x^2 + 7x}{3x}\)

Denominators: \(3x + 21 = 0 \implies x = -7\); \(3x = 0 \implies x = 0\).
Divisor numerator: \(x^2 + 7x = 0 \implies x(x + 7) = 0 \implies x = 0, -7\).
Excluded values: \(\{-7, 0\}\).

• Option 2: \(\frac{7x}{x^2 - 10x + 21} \div \frac{x + 7}{7}\)

Denominators: \(x^2 - 10x + 21 = 0 \implies (x - 3)(x - 7) = 0 \implies x = 3, 7\).
Divisor numerator: \(x + 7 = 0 \implies x = -7\).
Excluded values: \(\{-7, 3, 7\}\).

• Option 3: \(\frac{x + 7}{x^2 + 6x - 7} \div \frac{7}{2x + 14}\)

Denominators: \(x^2 + 6x - 7 = 0 \implies (x + 7)(x - 1) = 0 \implies x = -7, 1\); \(2x + 14 = 0 \implies x = -7\).
Divisor numerator: \(7
eq 0\).
Excluded values: \(\{-7, 1\}\).

• Option 4: \(\frac{x - 7}{x^2 + 4x - 21} \div \frac{x^2 + 49}{x + 7}\)

Denominators: \(x^2 + 4x - 21 = 0 \implies (x + 7)(x - 3) = 0 \implies x = -7, 3\); \(x + 7 = 0 \implies x = -7\).
Divisor numerator: \(x^2 + 49 = 0 \implies \text{no real solutions}\).
Excluded values: \(\{-7, 3\}\).

Select the option containing \(x = 7\)

Only Option 2 has \(x = 7\) as an excluded value because it makes the denominator \(x^2 - 10x + 21\) equal to zero.

Answer:

• (A) \(\frac{x^2 - 49}{3x + 21} \div \frac{x^2 + 7x}{3x}\)
• (B) \(\frac{7x}{x^2 - 10x + 21} \div \frac{x + 7}{7}\) (Correct answer)
• (C) \(\frac{x + 7}{x^2 + 6x - 7} \div \frac{7}{2x + 14}\)
• (D) \(\frac{x - 7}{x^2 + 4x - 21} \div \frac{x^2 + 49}{x + 7}\)