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

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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:
  $$\frac{4}{x+2}=\frac{16}{x+5},x \in \left(-\infty,-5ight)\cup \left(-5,-2ight)\cup \left(-2,+\inftyight)$$
- step2: Cross multiply:
  $$4\left(x+5ight)=\left(x+2ight)	imes 16$$
- step3: Simplify the equation:
  $$4\left(x+5ight)=16\left(x+2ight)$$
- step4: Rewrite the expression:
  $$4\left(x+5ight)=4	imes 4\left(x+2ight)$$
- step5: Evaluate:
  $$x+5=4\left(x+2ight)$$
- 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:
  $$x=-1,x \in \left(-\infty,-5ight)\cup \left(-5,-2ight)\cup \left(-2,+\inftyight)$$
- step14: Find the intersection:
  $$x=-1$$
The solution to the equation $$ \frac{4}{x+2}=\frac{16}{x+5} $$ is $$ x=-1 $$.
$$ x=-1 $$

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