Question

solve.
\\(\frac{4}{x + 4} = \frac{6}{x}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution(s) is/are \\(\square\\).
(type an integer or a simplified fraction. use a comma to separate answers as needed.)

b. the solution set is \\(\\{x \mid x\\) is a real number and \\(x \
eq \square\\}\\).
(type an integer or a simplified fraction. use a comma to separate answers as needed.)

c. there is no solution.

Explanation:

Step1: Cross - multiply to eliminate fractions

Given the equation \(\frac{4}{x + 4}=\frac{6}{x}\), cross - multiplying (which is valid when \(x
eq0\) and \(x
eq - 4\)) gives us \(4x=6(x + 4)\).

Step2: Expand the right - hand side

Expand \(6(x + 4)\) using the distributive property \(a(b + c)=ab+ac\). So, \(6(x + 4)=6x+24\). The equation becomes \(4x = 6x+24\).

Step3: Solve for \(x\)

Subtract \(6x\) from both sides of the equation: \(4x-6x=6x + 24-6x\), which simplifies to \(-2x=24\). Then divide both sides by \(-2\): \(x=\frac{24}{-2}=- 12\).
We need to check if \(x = - 12\) is a valid solution. Substitute \(x=-12\) into the original equation's denominators: \(x=-12
eq0\) and \(x + 4=-12 + 4=-8
eq0\). So, \(x=-12\) is a valid solution.

Answer:

A. The solution(s) is/are \(-12\).