Solve the inequality. Write the solution set in interval notation if possible. Simplify your answer if necessary. $$ 6 y(2 y-3)^{4}(4 y+1)^{5}(y-4)

Question

Solve the inequality. Write the solution set in interval notation if possible. Simplify your answer if necessary. $$ 6 y(2 y-3)^{4}(4 y+1)^{5}(y-4)<0 $$ The solution set is $$ \square $$ .

UpStudy AI · Accepted Answer

Solve the equation $$ 6y(2y-3)^4(4y+1)^5(y-4)<0 $$.
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
  $$6y\left(2y-3ight)^{4}\left(4y+1ight)^{5}\left(y-4ight)<0$$
- step1: Rewrite the expression:
  $$6y\left(2y-3ight)^{4}\left(4y+1ight)^{5}\left(y-4ight)=0$$
- step2: Elimination the left coefficient:
  $$y\left(2y-3ight)^{4}\left(4y+1ight)^{5}\left(y-4ight)=0$$
- step3: Separate into possible cases:
  $$\begin{align}&y=0\&\left(2y-3ight)^{4}=0\&\left(4y+1ight)^{5}=0\&y-4=0\end{align}$$
- step4: Solve the equation:
  $$\begin{align}&y=0\&y=\frac{3}{2}\&y=-\frac{1}{4}\&y=4\end{align}$$
- step5: Determine the test intervals:
  $$\begin{align}&y<-\frac{1}{4}\&-\frac{1}{4}4\end{align}$$
- step6: Choose a value:
  $$\begin{align}&y_{1}=-1\&y_{2}=-\frac{1}{8}\&y_{3}=1\&y_{4}=3\&y_{5}=5\end{align}$$
- step7: Test the chosen value:
  $$\begin{align}&y<-\frac{1}{4}	extrm{ }	extrm{is the solution}\&-\frac{1}{4}4	extrm{ }	extrm{is not a solution}\end{align}$$
- step8: The final solution is $$y \in \left(-\infty,-\frac{1}{4}ight)\cup \left(0,\frac{3}{2}ight)\cup \left(\frac{3}{2},4ight):$$
  $$y \in \left(-\infty,-\frac{1}{4}ight)\cup \left(0,\frac{3}{2}ight)\cup \left(\frac{3}{2},4ight)$$
The solution set for the inequality $$6y(2y-3)^4(4y+1)^5(y-4)<0$$ is $$y \in (-\infty,-\frac{1}{4}) \cup (0,\frac{3}{2}) \cup (\frac{3}{2},4)$$.
The solution set is $$ y \in (-\infty, -\frac{1}{4}) \cup (0, \frac{3}{2}) \cup (\frac{3}{2}, 4) $$.

Solve the inequality. Write the solution set in interval notation if possible. Simplify your answer if necessary.

The solution set is .

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Solve the inequality. Write the solution set in interval notation if possible. Simplify your answer if necessary. The solution set is .