MathLabs

Problem 4

Show that the solution set of the inequality ∑k=170kx−k≥54\sum_{k=1}^{70}\frac{k}{x-k}\ge\frac{5}{4} is a union of disjoint intervals whose total length is 19881988.
Step 3 of 4: Express the total length through the roots
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L=∑n=170(rn−n)=∑n=170rn−35⋅71L=\sum_{n=1}^{70}(r_n-n)=\sum_{n=1}^{70}r_n-35\cdot71
Detailed analysis

The total length is L=∑n=170(rn−n)=∑n=170rn−35⋅71L=\sum_{n=1}^{70}(r_n-n)=\sum_{n=1}^{70}r_n-35\cdot71. Clear denominators in f(x)=5/4f(x)=5/4; the resulting degree-70 polynomial has roots r1,…,r70r_1,\ldots,r_{70}.