CAT Practice : Inequalities

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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?

Integer Roots - Trial and Error

    Q.4: How many integer values of x satisfy the inequality x( x + 2)(x + 4)(x + 6) < 200.

 

  • Correct Answer
    There are a total of nine values

Detailed Solution

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.
Correct Answer : 9 different values.

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Inequalities are crucial to understand many topics that are tested in the CAT. Having a good foundation in this subject will make us tackling questions in Coordinate Geometry, Functions, and most importantly in Algebra much more comfortable.