Question

solving rational equations

solve the following rational equation.

\\\frac{x + 5}{2x} = \frac{3}{x - 6}\\

\\(x = \\) ? ,

enter the smallest solution first.

Explanation:

Set up the equation and cross-multiply

$$ \frac{x+5}{2x} = \frac{3}{x-6} $$

$$ (x+5)(x-6) = 3(2x) $$

Expand and simplify to standard quadratic form

$$ x^2 - x - 30 = 6x $$

$$ x^2 - 7x - 30 = 0 $$

Solve the quadratic equation by factoring

$$ (x-10)(x+3) = 0 $$

$$ x = 10 \quad \text{or} \quad x = -3 $$

Both values satisfy the domain restrictions \(x
eq 0\) and \(x
eq 6\).
The smallest solution is \(-3\) and the largest is \(10\).

Answer:

Solve the following rational equation.

$$\frac{x+5}{2x} = \frac{3}{x-6}$$

\(x =\) <blank>-3</blank>, <blank>10</blank>
Enter the smallest solution first.