EgmatQuantExpert
Find the product of the integer values of x that satisfy the inequality \(\frac{(x^2-4)^3}{(x-5)^5 (x^2 - 9)^4} > 0\)
We could solve this question easily with the tool called:
factor table with sign\(\frac{(x^2-4)^3}{(x-5)^5 (x^2 - 9)^4} > 0 \iff \frac{(x-2)^3(x+2)^3}{(x-5)^5(x-3)^4(x+3)^4} > 0\)
First, find the value that all factors are equal to 0
\(\begin{split}
x-2=0 &\iff x=2 \\
x+2=0 &\iff x=-2\\
x-5=0 &\iff x=5\\
x-3=0 &\iff x=3\\
x+3=0 &\iff x=-3
\end{split}\)
Now, make a factor table with sign like this
Attachment:
Capture table 1.PNG [ 12.77 KiB | Viewed 17653 times ]
In line (1), we review all possible values of \(x\) that make each fator equal to 0. The order is ascending (from lowest to highest)
Next, from line (2) to (6), we review the sign of each raw factor without power that based on the range value of \(x\). Character "|" means that we no need to care about the specific value of them, just the sign positive (+) or negative (-).
Next, from line (7) to (11), we review the sign of each fator with power that exists in the expression.
Finally, in line (12), we could quickly review the sign of the expression.
Hence, we have \(\frac{(x-2)^3(x+2)^3}{(x-5)^5(x-3)^4(x+3)^4} > 0 \iff x \in (-2,2) \cup (5, +\infty) \)
You could find more about this tool in this link:
fator-table-with-sign-the-useful-tool-to-solve-polynomial-inequalities-229988.html#p1771368