For the function $$ h(x)=\frac{4 x}{(x-3)(x+5)} $$, solve the following inequality. $$ h(x)

Question

For the function $$ h(x)=\frac{4 x}{(x-3)(x+5)} $$, solve the following inequality. $$ h(x)<0 $$

UpStudy AI · Accepted Answer

Solve the equation $$ \frac{4x}{(x-3)(x+5)}<0 $$.
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
  $$\frac{4x}{\left(x-3ight)\left(x+5ight)}\lt 0$$
- step1: Find the domain:
  $$\frac{4x}{\left(x-3ight)\left(x+5ight)}\lt 0,x \in \left(-\infty,-5ight)\cup \left(-5,3ight)\cup \left(3,+\inftyight)$$
- step2: Set the numerator and denominator of $$\frac{4x}{\left(x-3ight)\left(x+5ight)}$$ equal to 0$$:$$
  $$\begin{align}&4x=0\&\left(x-3ight)\left(x+5ight)=0\end{align}$$
- step3: Calculate:
  $$\begin{align}&x=0\&x=3\&x=-5\end{align}$$
- step4: Determine the test intervals:
  $$\begin{align}&x<-5\&-53\end{align}$$
- step5: Choose a value:
  $$\begin{align}&x_{1}=-6\&x_{2}=-3\&x_{3}=2\&x_{4}=4\end{align}$$
- step6: Test the chosen value:
  $$\begin{align}&x<-5	extrm{ }	extrm{is the solution}\&-53	extrm{ }	extrm{is not a solution}\end{align}$$
- step7: The final solution is $$x \in \left(-\infty,-5ight)\cup \left(0,3ight):$$
  $$x \in \left(-\infty,-5ight)\cup \left(0,3ight)$$
- step8: Check if the solution is in the defined range:
  $$x \in \left(-\infty,-5ight)\cup \left(0,3ight),x \in \left(-\infty,-5ight)\cup \left(-5,3ight)\cup \left(3,+\inftyight)$$
- step9: Find the intersection:
  $$x \in \left(-\infty,-5ight)\cup \left(0,3ight)$$
The solution to the inequality $$ h(x) < 0 $$ is $$ x \in (-\infty, -5) \cup (0, 3) $$.
The solution to $$ h(x) < 0 $$ is $$ x $$ in the intervals $$ (-\infty, -5) $$ and $$ (0, 3) $$.

For the function , solve the following inequality.

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