Problem 4

Show that the solution set of the inequality

\sum_{k=1}^{70}\frac{k}{x-k}\ge\frac{5}{4}

is a union of disjoint intervals whose total length is

1988

Step 2 of 4: Locate the final interval

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r_{70}>70,\qquad f(r_{70})=\frac{5}{4}

Detailed analysis

For $x<1$ , every term is negative, so the inequality fails. On $(70,+\infty)$ , $f$ decreases from $+\infty$ to $0$ , so there is a unique $r_{70}>70$ with $f(r_{70})=5/4$ . Therefore the solution set is exactly the disjoint union of $(n,r_n]$ for $1\le n\le70$ .