5. Rezolvați in R inecuația: $$ \frac{(3+2 x)(x-4)}{x-2}

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5. Rezolvați in R inecuația: $$ \frac{(3+2 x)(x-4)}{x-2}<0 $$

UpStudy AI · Accepted Answer

It seems there was an issue with the input format. Let's try to solve the inequality $$ \frac{(3+2x)(x-4)}{x-2}<0 $$ in R.

To solve the inequality, we need to find the values of $$ x $$ for which the expression $$ \frac{(3+2x)(x-4)}{x-2} $$ is less than 0.

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

5. Rezolvați in R inecuația: \( \frac{(3+2 x)(x-4)}{x-2}<0 \)

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