Question

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

Explanation:

Step1: Find the roots

Set \(-5(x - 4)^{5}(x - 5)=0\). Then \(x-4 = 0\) gives \(x = 4\) and \(x - 5=0\) gives \(x=5\).

Step2: Analyze the sign

Consider the intervals \((-\infty,4)\), \((4,5)\) and \((5,\infty)\).
For \(x<4\), let \(x = 3\). Then \(-5(3 - 4)^{5}(3 - 5)=-5\times(-1)^{5}\times(-2)=- 10<0\), the inequality holds.
For \(4For \(x>5\), let \(x = 6\). Then \(-5(6 - 4)^{5}(6 - 5)=-5\times2^{5}\times1=-160<0\), the inequality holds.

Answer:

The solution of the inequality \(-5(x - 4)^{5}(x - 5)\leq0\) is \(x\leq4\) or \(x\geq5\). On the number - line, we plot closed - circles at \(x = 4\) and \(x = 5\) (since the inequality includes equality). Then we draw a ray to the left of \(x = 4\) and a ray to the right of \(x = 5\).