$3$ . Solve the rational equation: $\newline$ (a) $\frac{x}{x+3}=\frac{8}{x+6}$

Full solution

3

. Solve the rational equation:

\newline

(a)

\frac{x}{x+3}=\frac{8}{x+6}

Cross-Multiply Fractions: Cross-multiply to eliminate the fractions. $\newline$ Set up the equation $\frac{x}{x+3}=\frac{8}{x+6}$ and cross-multiply to get $x(x+6) = 8(x+3)$ .
Distribute and Expand: Distribute and expand both sides of the equation. $x(x+6) = 8(x+3)$ becomes $x^2 + 6x = 8x + 24$ .
Move Terms to One Side: Move all terms to one side to set the equation to zero. Subtract $8x$ and $24$ from both sides to get $x^2 + 6x - 8x - 24 = 0$ , which simplifies to $x^2 - 2x - 24 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ Look for two numbers that multiply to $-24$ and add to $-2$ . The numbers $-6$ and $+4$ work, so factor the equation as $(x - 6)(x + 4) = 0$ .
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ Set $x - 6 = 0$ , which gives $x = 6$ . $\newline$ Set $x + 4 = 0$ , which gives $x = -4$ .
Check for Extraneous Solutions: Check for extraneous solutions by substituting the values back into the original equation. $\newline$ For $x = 6$ : $\frac{6}{6+3} = \frac{8}{6+6}$ simplifies to $\frac{2}{3} = \frac{2}{3}$ , which is true. $\newline$ For $x = -4$ : $\frac{-4}{-4+3} = \frac{8}{-4+6}$ simplifies to $\frac{-4}{-1} = \frac{8}{2}$ , which is false because $-4 \neq 4$ . $\newline$ Therefore, $x = -4$ is an extraneous solution.
State Final Answer: State the final answer. $\newline$ The solution to the equation $\frac{x}{x+3}=\frac{8}{x+6}$ is $x = 6$ .