CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

The question is about finding No. of solutions of an inequality. We need to find out the number of possible integers that satisfy a given inequality. Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 1: How many positive integer values can x take that satisfy the inequality (x - 8) (x - 10) (x - 12).......(x - 100) < 0?

1. 25
2. 30
3. 35
4. 40

Video Explanation

Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : This is an excellent question to understand how a polynomial behaves when the variable 'x' takes values between roots. Think about how you can make the polynomial take positive and negative values. What values of 'x' should you substitute in the polynomial?

Let us try out a few values to see if that gives us anything.
When x = 8, 10, 12, ....100 this goes to zero. So, these cannot be counted.
When x = 101, 102 or beyond, all the terms are positive, so the product will be positive.
So, straight-away we are down to numbers 1, 2, 3, ...7 and then odd numbers from there to 99.

Let us substitute x =1,
All the individual terms are negative. There are totally 47 terms in this list (How? Figure that out). Product of 47 negative terms will be negative.
So, x = 1 works. So, will x =2, 3, 4, 5, 6, and 7.

Remember, product of an odd number of negative terms is negative; product of even number of negative terms is positive. Now, this idea sets up the rest of the question.

When x = 9, there is one positive terms and 46 negative terms. So, the product will be positive.
When x = 11, there are two positive terms and 45 negative terms. So, the product will be negative.
When x = 13, there are three positive terms and 44 negative terms. So, the product will be positive.
and so on.

Essentially, alternate odd numbers need to be counted, starting from 11.
So, the numbers that will work for this inequality are 1, 2, 3, 4, 5, 6, 7...and then 11, 15, 19, 23, 27, 31,..... and so on.
What will be the last term on this list?
99, because when x = 99, there are 46 positive terms and 1 negative term.
So, we need to figure out how many terms are there in the list 11, 15, 19,....99. These can be written as

4 * 2 + 3

4 * 3 + 3
4 * 4 + 3
4 * 5 + 3
4 * 6 + 3
...
4 * 24 + 3
A set of 23 terms. So, total number of values = 23 + 7 = 30. 30 positive integer values of x exist satisfying the condition.

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

Hence the answer is "30"

Choice B is the correct answer.