24. \( \frac{4}{x+2}=\frac{16}{x+5} \)

Solve the equation \( \frac{4}{x+2}=\frac{16}{x+5} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{4}{x+2}=\frac{16}{x+5}\) - step1: Find the domain: \(\frac{4}{x+2}=\frac{16}{x+5},x \in \left(-\infty,-5\right)\cup \left(-5,-2\right)\cup \left(-2,+\infty\right)\) - step2: Cross multiply: \(4\left(x+5\right)=\left(x+2\right)\times 16\) - step3: Simplify the equation: \(4\left(x+5\right)=16\left(x+2\right)\) - step4: Rewrite the expression: \(4\left(x+5\right)=4\times 4\left(x+2\right)\) - step5: Evaluate: \(x+5=4\left(x+2\right)\) - step6: Expand the expression: \(x+5=4x+8\) - step7: Move the expression to the left side: \(x-4x=8-5\) - step8: Add and subtract: \(-3x=8-5\) - step9: Add and subtract: \(-3x=3\) - step10: Change the signs: \(3x=-3\) - step11: Divide both sides: \(\frac{3x}{3}=\frac{-3}{3}\) - step12: Divide the numbers: \(x=-1\) - step13: Check if the solution is in the defined range: \(x=-1,x \in \left(-\infty,-5\right)\cup \left(-5,-2\right)\cup \left(-2,+\infty\right)\) - step14: Find the intersection: \(x=-1\) The solution to the equation \( \frac{4}{x+2}=\frac{16}{x+5} \) is \( x=-1 \).