Question 92

# For how many integers n, will the inequality $$(n - 5) (n - 10) - 3(n - 2)\leq0$$ be satisfied?

Solution

$$(n - 5) (n - 10) - 3(n - 2)\leq0$$
=> $$n^2 - 15n + 50 - 3n + 6 \leq 0$$
=> $$n^2 - 18n + 56 \leq 0$$
=> $$(n - 4)(n - 14) \leq 0$$
=> Thus, n can take values from 4 to 14. Hence, the required number of values are 14 - 4 + 1 = 11.