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CAT 2017 Slot 1 — QA Question 26

Solving InequalitiesEasy

For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied?

Answer & solution

Correct answer: 11

Answer: 11

Solution

Easy

Expand to a quadratic in nn, factor it, and find where it is ≤0\le 0. The integers in that closed interval are the answer.

1

Expand and collect terms.

(n−5)(n−10)−3(n−2)≤0⇒ n2−15n+50−3n+6≤0⇒ n2−18n+56≤0\begin{aligned} &(n-5)(n-10) - 3(n-2) \le 0\\ &\Rightarrow\ n^2 - 15n + 50 - 3n + 6 \le 0\\ &\Rightarrow\ n^2 - 18n + 56 \le 0 \end{aligned}
2

Factor the quadratic. We need two numbers multiplying to 5656 and adding to 1818: namely 44 and 1414.

(n−4)(n−14)≤0⇒ 4≤n≤14(product of two factors ≤0)\begin{aligned} &(n-4)(n-14) \le 0\\ &\Rightarrow\ 4 \le n \le 14 \quad\text{(product of two factors }\le 0\text{)} \end{aligned}
3

Count the integers. The integers 4,5,6,…,144,5,6,\dots,14 satisfy the inequality.

14−4+1=11\begin{aligned} &14 - 4 + 1 = 11 \end{aligned}
11 integers11 \text{ integers}

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CAT 2017 Slot 1 QA Q26: For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be sat — Solution | TheCATExam