Question

solving rational equations

solve the following rational equation.

\\(\frac{x}{x + 3} = \frac{8}{x - 2}\\)

\\(x = ?, \box\\)

enter the smallest solution first

Explanation:

Set up the equation and identify restrictions

$$ \frac{x}{x+3} = \frac{8}{x-2} \quad (x eq -3, x eq 2) $$

Cross-multiply to clear fractions

$$
LATEXBLOCK0
$$

Rearrange into standard quadratic form and solve

$$
LATEXBLOCK1
$$

Both solutions satisfy the domain restrictions \(x
eq -3, 2\).
The smallest solution is \(-2\), and the largest is \(12\).

Answer:

Solve the following rational equation.

$$\frac{x}{x+3} = \frac{8}{x-2}$$

\(x =\) <blank>\(-2\)</blank>, <blank>\(12\)</blank>
Enter the smallest solution first.