How many positive integer values can x take that satisfy the inequalit

Question

How many positive integer values can x take that satisfy the inequality  (x - 8)(x - 10)(x - 12)...(x - 100) < 0 ?

A. 25
B. 30
C. 35
D. 40
E. 47

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Knightclaw · Answer

using wavy curve, we can see that the sign of the inequality is going to alternate after each solution
so total number of terms is 47. sign will alternate at each of them . so 23 of the integers will result in < 0
also all positive integers less than 8 will also yield < 0
hence 23 + 7 =30
answer is B

JerryAtDreamScore · Answer

Breaking Down the Info:

There are  \frac{100 - 8}{2} + 1 = 47  factors in this product. Then our goal is to have an odd number of negative terms.

We can start by putting in x = 99, this gives us 46 positive terms and 1 negative term. Next, we can try x = 97 but that would give us 2 negative terms and a positive product. After that, try x = 95 to get 3 negative terms and a negative product. Thus, we can keep going by subtracting 4's.

Therefore, we are allowed to take x = 99, 95, 91, .... 11, and then 7, 6, 5, 4, 3, 2, 1 since we have 47 negative terms when x is small enough.

There are  \frac{99 - 11}{4} + 1 = 23  integers from 11 to 99, and 7 more small integers so that is  30  integers in total.

Answer: B

BrushMyQuant · Answer

We need to find how many positive integer values can x take that satisfy the inequality \((x - 8)(x - 10)(x - 12)...(x - 100) < 0\)

(x - 8)(x - 10)(x - 12)...(x - 100) < 0
=> To find the terms we can just find how many numbers are in the series 8, 10, 12, ..., 100

Now, this is an Arithmetic series with first term 8 and last term 100 and common difference = 2
=> Number of terms, n = \(\frac{Last Term - First Term }{ Common Difference}\) + 1 = \(\frac{100 - 8}{2}\) + 1 = \(\frac{92}{2}\) + 1 = 47 terms
( Watch this video to learn about the Basics of Arithmetic Series )

Now, product of 47 terms has to be < 0
=> We need to have

Now, x cannot be > 100 as this will make the entire expression > 0
=> NOT POSSIBLE

Now, x cannot be equal to 8, 10, ...., 100 as this will make the entire expression 0
=> NOT POSSIBLE

Now, 1 ≤ x ≤ 7 will make all terms as negative
=> POSSIBLE
=> 7 values ( 1,2,...7)

Now, if x = 9, 13, 17, ... then we will have odd number of negative terms and even number of positive terms
=> Entire expression < 0
=> POSSIBLE
=> \(\frac{47-1}{2}\) = 23 terms

Now, if x = 11, 15, 19, ... then we will have even number of negative terms and odd number of positive terms
=> Entire expression > 0
=> NOT POSSIBLE

=> Total values of values of x = 7 + 23 = 30 values

So, Answer will be B
Hope it helps!

Watch the following video to learn the Basics of Inequalities

Pjuuuuuuuu · Answer

Hey,

I thought the wrong way but not sure what’s wrong.

I thought we could get even number of negative + one positive to get negative.
Then the number could be 9,11,13,15,17,19, 21,23,25,27,29,31,33,35,37and for the last one instead of having 101, we have 99-100=-1, so the number would be (101-9)/2=46, not adding one because the last one is using 99 not 101. In this way I got 46. Can anyone help?

Bunuel · Answer

For the product of 47 terms to be negative, an  odd number  of those terms must be negative and the remaining terms must be positive.

The numbers like 9, 13, 17, ..., 97 do not work, so you're left with 11, 15, ..., 99, which gives 23 numbers. Adding the seven numbers 1 through 7 gives a total of 30 numbers.

Check the solutions above for more.

Kinshook · Answer

How many positive integer values can x take that satisfy the inequality  (x - 8)(x - 10)(x - 12)...(x - 100) < 0 ?

Let  E = (x - 8)(x - 10)(x - 12)...(x - 100) 
Number of terms = (100 - 8)/2 + 1 = 47

x < 8; x = {1,2,3,4,5,6,7}: E < 0
x = {11,15,19, .....99}: E< 0

Total number of positive integer values can x take that satisfy the inequality  (x - 8)(x - 10)(x - 12)...(x - 100) < 0  = 7 + (99-11)/4+1 = 8 + 22 = 30

IMO B

How many positive integer values can x take that satisfy the inequalit

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GMAT Problem Solving (PS) Questions

How many positive integer values can x take that satisfy the inequalit

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