Question

solve for x and graph the solution.
-4(x - 1)^5(x - 5)^3 < 0
plot the endpoints. select an endpoint to change it from a segment, ray, or line to delete it.

Explanation:

Step1: Analyze the sign of the leading - factor

The leading - factor is $-4$, which is negative.

Step2: Find the roots of the polynomial

Set $(x - 1)^5(x - 5)^3=0$. The roots are $x = 1$ and $x = 5$ (since if $(x - 1)^5=0$ then $x = 1$ and if $(x - 5)^3=0$ then $x = 5$).

Step3: Use the sign - chart method

Consider the intervals $(-\infty,1)$, $(1,5)$ and $(5,\infty)$.
For $x\lt1$, let's take $x = 0$. Then $-4(0 - 1)^5(0 - 5)^3=-4\times(-1)\times(-125)=-500\lt0$.
For $1\lt x\lt5$, let's take $x = 2$. Then $-4(2 - 1)^5(2 - 5)^3=-4\times1\times(-27)=108\gt0$.
For $x\gt5$, let's take $x = 6$. Then $-4(6 - 1)^5(6 - 5)^3=-4\times3125\times1=-12500\lt0$.

Answer:

The solution of the inequality $-4(x - 1)^5(x - 5)^3\lt0$ is $x\lt1$ or $x\gt5$. On the number - line, we plot open circles at $x = 1$ and $x = 5$ (because the inequality is strict, i.e., $\lt$ not $\leq$), and shade the intervals $(-\infty,1)$ and $(5,\infty)$.