Question

solve for all values of x. \\(\frac{x + 4}{x + 8} = \frac{6}{x}\\) answer attempt 1 out of 2 \\(\circ\\) additional solution \\(\circ\\) no solution \\(x = \square\\) submit answer

Explanation:

Step1: Cross - multiply to eliminate fractions

Given the equation \(\frac{x + 4}{x+8}=\frac{6}{x}\), cross - multiplying (which is valid when \(x
eq0\) and \(x
eq - 8\)) gives us \(x(x + 4)=6(x + 8)\).

Step2: Expand both sides

Expanding the left - hand side: \(x(x + 4)=x^{2}+4x\).
Expanding the right - hand side: \(6(x + 8)=6x + 48\).
So the equation becomes \(x^{2}+4x=6x + 48\).

Step3: Rearrange into standard quadratic form

Subtract \(6x\) and \(48\) from both sides to get \(x^{2}+4x-6x - 48 = 0\).
Simplify the left - hand side: \(x^{2}-2x - 48=0\).

Step4: Factor the quadratic equation

We need to find two numbers that multiply to \(- 48\) and add up to \(-2\). The numbers are \(-8\) and \(6\).
So, \(x^{2}-2x - 48=(x - 8)(x+6)=0\).

Step5: Solve for \(x\)

Set each factor equal to zero:

• If \(x - 8=0\), then \(x = 8\).
• If \(x+6=0\), then \(x=-6\).

We need to check these solutions in the original equation:

• For \(x = 8\): Left - hand side \(\frac{8 + 4}{8+8}=\frac{12}{16}=\frac{3}{4}\), Right - hand side \(\frac{6}{8}=\frac{3}{4}\). So \(x = 8\) is a valid solution.
• For \(x=-6\): Left - hand side \(\frac{-6 + 4}{-6+8}=\frac{-2}{2}=-1\), Right - hand side \(\frac{6}{-6}=-1\). So \(x=-6\) is a valid solution.

Answer:

\(x = 8\) or \(x=-6\)