CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

CAT Questions | Algebra | Inequalities - Integer Roots

The question is about integer roots. Our task is to find out the number of integers which satisfies the 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 4: How many integer values of x satisfy the inequality x( x + 2)(x + 4)(x + 6) < 200?


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Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : Expanding and factorising again would be too time-consuming. Try substituting some extreme values and see how the equation behaves. When will it be negative?

To begin with - 0, -2, -4 and -6 work. These are the values for which the left-hand side goes to zero.

There are 4 terms in the product. If all 4 are positive or all 4 are negative the product will be positive.

The product can be negative only if exactly 1 or exactly 3 are negative. When 1 or 3 terms are negative, the product is clearly less than 200.

When x = -1, one term is negative
When x = -5, three terms are negative
So, adding these two numbers also to the set of solutions {-6, -5, -4, -2, -1, 0} satisfy the inequality.
Beyond this it is just trial and error.

Let us try x = 1. Product is 1 * 3 * 5 * 7 = 105. This works
x = -7 gives the same product. So, that also works.

So, the solution set is now refined to {-7, -6, -5, -4, -2, -1, 0, 1}

x = 2 => Product is 2 * 4 * 6 * 8 = 8 * 48. Not possible. Any x greater than 1 does not work.
x = -8 is also not possible. Any value of x less than -7 does not work.

So, the solution set stays as {-7, -6, -5, -4, -2, -1, 0, 1}

The one missing value in this sequence is -3. When x = -3, product becomes -3 * -1 * 1 * 3. = 9. This also holds good.

So, values {-7, -6,-5, -4, -3, -2, -1, 0, 1} hold good. 9 different values satisfy this inequality.

The question is "How many integer values of x satisfy the inequality x( x + 2)(x + 4)(x + 6) < 200?"

Hence the answer is "There are a total of nine values "


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