This GMAT quant practice question is a problem solving question from Inequalities. Concept: Inequality of algebraic expressions. A GMAT 650 quantitative reasoning sample question.

Question 3: What is the smallest integer that satisfies the inequality \\frac{x - 3}{x^2 - 8x - 20}\\) > 0?

- -2
- 10
- 3
- -1
- 0

The smallest integer that satisfies the inequality is -1.

@ INR

Let us factorize the denominator and rewrite the expression as \\frac{x - 3}{(x - 10)(x + 2)}\\) > 0

**Approach:** Equate each of the terms of the expression to zero to identify the values of x in which the inequality holds good.

The values that are relevant to us are x = 3, x = 10 and x = -2.

Let us arrange these values in ascending order: -2, 3 and 10.

The quickest way to solve inequalities questions after arriving at these values is verifying whether the inequality holds good at the following intervals.

**Interval 1:** x < -2.

Pick a value in that range and check whether the inequality holds good.

Let us take x = -10. When x = -10, the value of \\frac{x - 3}{(x - 10)(x + 2)}\\) is \\frac{-10 - 3}{(-10 - 10)(-10 + 2)}\\).

The value of the expression in this interval is negative; the inequality __DOES NOT__ hold good in this interval.

**Interval 2:** -2 < x < 3.

Let us take x = 0. When x = 0, \\frac{x - 3}{(x - 10)(x + 2)}\\) = \\frac{0 - 3}{(0 - 10)(0 + 2)}\\) > 0; the inequality holds good in this interval.

We found that the inequality holds good in the interval -2 < x < 3

The least integer value that x can take in the interval -2 < x < 3 is x = -1.

So, the correct answer is -1.

__Remember__: We have to find out the least integer value. And we have arrived at -1.

Do not waste time computing the entire range of values of x that satisfy the inequality.

**Note**: In any inequality question, when the question asks us to determine the intervals in which the inequality holds good, we have to eliminate values of x that will result in the denominator becoming zero.

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