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info@snapandsolve.comTo solve the equation \( \frac{x}{x+6} \div \frac{5x}{x+1} = \frac{2}{5(x+6)} \), first rewrite the division as multiplication by the reciprocal:
\[ \frac{x}{x+6} \times \frac{x+1}{5x} = \frac{2}{5(x+6)} \]
Multiply both sides by \( 5(x+6) \) to eliminate the denominator on the right:
\[ x(x+1) = 2 \]
Expand and simplify:
\[ x^2 + x = 2 \]
Rearrange to form a quadratic equation:
\[ x^2 + x - 2 = 0 \]
Factor the quadratic equation:
\[ (x - 1)(x + 2) = 0 \]
Set each factor to zero and solve for \( x \):
Check for extraneous solutions by substituting back into the original equation. \( x = -6 \) is not valid as it makes the denominator zero. \( x = 1 \) is a valid solution.
Answer: a. \( x = 1 \)
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