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Bunuel
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How many integers n satisfy (n 5) (n 10) 3(n 2) 0 ? 22 Apr 2022, 02:45
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

65% (hard)

Question Stats:

60% (02:26) correct 40% (02:41) wrong based on 186 sessions

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How many integers n satisfy \((n – 5) (n – 10) – 3(n – 2) ≤ 0\) ?

A. 8
B. 9
C. 10
D. 11
E. 12




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D
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How many integers n satisfy (n 5) (n 10) 3(n 2) 0 ? 29 Apr 2022, 04:02
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Bunuel
How many integers n satisfy \((n – 5) (n – 10) – 3(n – 2) ≤ 0\) ?

A. 8
B. 9
C. 10
D. 11
E. 12




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Solving the eqn, we get (n-4) * (n-14) <= 0

n will take integral values from 4 to 14

Total 11 values

Option D
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How many integers n satisfy (n 5) (n 10) 3(n 2) 0 ? 03 Mar 2023, 03:48
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Bunuel
How many integers n satisfy \((n – 5) (n – 10) – 3(n – 2) ≤ 0\) ?

A. 8
B. 9
C. 10
D. 11
E. 12
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Solution:

  • We are given \((n – 5) (n – 10) – 3(n – 2) ≤ 0\)
    \(⇒n^2-10n-5n+50-3n+6≤ 0\)
    \(⇒n^2-18n+56 \le 0\)
    \(⇒n^2-4n-14n+56 \le 0\)
    \(⇒n(n-4)-14(n-4)\le 0\)
    \(⇒(n-4)(n-14)\le 0\)
  • So, the possible integral values of n will be from 4 to 14 = 14 - 4 + 1 = 11

Hence the right answer is Option D
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How many integers n satisfy (n 5) (n 10) 3(n 2) 0 ? 05 Mar 2023, 09:59
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We need to find how many integers n satisfy \((n – 5) (n – 10) – 3(n – 2) ≤ 0\) ?

\((n – 5) (n – 10) – 3(n – 2) ≤ 0\)
=> \(n^2\) -5n - 10n + 50 -3n + 6 ≤ 0
=> \(n^2\) -18n + 56 ≤ 0
=> \(n^2\) -14n -4n + 56 ≤ 0
=> n*(n - 14) - 4(n - 14) ≤ 0
=> (n - 14) * (n - 4) ≤ 0

Two ways to solve this now

Method 1: Logic

(n - 14) * (n - 4) ≤ 0
Product of two numbers is ≤ 0
=> Either one is positive and other is negative or at least one is 0
0 case can come when n = 4 or 14

Lets solve for Either to be negative
If n < 4 then both (n - 4) and (n - 14) will be negative => NOT POSSIBLE
If n > 14 then both (n - 4) and (n - 14) will be positive => NOT POSSIBLE
If 4 ≤ n ≤ 14 then (n - 14) * (n - 4) ≤ 0
=> Possible values are from 4 to 14 inclusive
=> 14 - 4 + 1 = 11 values

Method 2: Sine Wave / Curvy Method

We will plot 4 and 14 on the number line and put a filled dot on those two numbers to indicate that both are included. Refer below image:

Attachment:
4 to 14.jpg
4 to 14.jpg [ 13.27 KiB | Viewed 1905 times ]

Since the inequality sign is ≤ so we will take the values in "-" range

=> 4 ≤ n ≤ 14
=> 11 integer values are possible.

So, Answer will be D
Hope it helps!

Watch the following video to learn How to Solve Inequality Problems

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How many integers n satisfy (n 5) (n 10) 3(n 2) 0 ? 17 May 2025, 04:25
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