CAT Practice : Inequalities

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This one is similar to a previous question. DO NOT try to expand and solve the inequality with algebra. Try to substitute values systematically to see when the polynomial is less than 250.

Integers - Polynomials

Q.7: If x (x + 3) (x + 5) (x + 8) < 250, how many integer values can x take?

The inequality holds good for 11 terms. x can take all integer values from -9 to +1 (both inclusive).

Detailed Solution

Straight away we can see that x = 0 works. The product goes to zero in this case. When x takes values -3, -5 or -8, the product will go to zero and the inequality holds good.Now, let us substitute some other values.
When x = 1 => Product = 1 * 4 * 6 * 9 = 216 < 250 So, x = 1 holds good.
When x = 2 => The product is clearly greater than 250.

>So, thus far, we have seen that for x = 0, -3, -5, -8 or 1; the above inequality holds good.

There are 4 terms in this product. If all 4 are positive or all 4 are negative, the product will be positive. If exactly one term is positive or exactly one term is negative, the product will be negative.

Whenever the product is negative, the inequality will hold good. So, let us find the values of x for which the product will be negative.

x = -1 or -2, the product is negative, so the inequality will hold good.

Let us think of other values of x for which the product is negative. For the product to be negative, either 1 or 3 of the four terms should be negative.

When x is -6 or -7, three of the terms are negative and the product is negative.
So, for x = 0, 1, -3, -5, -8, -1, -2, -6 or -7 this holds good. We have seen 9 values thus far.
In ascending order, the values are -8, -7, -6, -5, -3, -2, -1, 0, 1

The value in between that we have thus far not verified is -4. Let us try -4 as well. In this case the product is -4 * -1 * 1 * 7 < 250

For x = -9, product is -9 * -6 * -4 * 1 = 216 < 250
Clearly for x = -10 or lesser the inequality does not hold good.

So, the inequality holds good for 11 terms. x can take all integers from -9 to +1 (both inclusive).

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More questions from Inequalities

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.