# GRE® Quant Practice | Algebra Q5

#### GRE Sample Questions | Inequalities of Quadratic Expressions

The GRE maths sample question given below is a word problem in Algebra. Solving inequalities of a quadratic expression is the concept tested in this GRE Quantitative Reasoning practice question. A medium difficulty question.

Question 5 : What is the least positive integer that will satisfy the inequality x2 + 17x – 84 > 0?

1. 1
2. 21
3. 4
4. 22
5. 5

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#### Step 1 of solving this GRE Inequalities Question: Factorise the given quadratic expression

x2 + 17x – 84 > 0
x2 + 21x – 4x – 84 > 0
x(x + 21) – 4 (x + 21) > 0
(x – 4)(x + 21) > 0

#### Step 2 of solving this GRE Inequalities Question: Identify Range to Evaluate

Based on the value of the roots of the quadratic equation, we have three ranges to check whether the inequality holds good.
The roots of the quadratic equation are x = 4 and x = −21.
The ranges to evaluate will therefore be x < -21; -21 < x < 4; and x > 4.

Key Result: If (x - a)(x - b) > 0, x will not lie between a and b.

Because the question is about finding the least positive integer value of x, we need not evaluate values of x < -21.

Range 1: -21 < x < 4
Even in this range, it will suffice if we substitute a positive value for x.
Check whether x = 1 will satisfy the inequality.
(1 - 4)(1 + 21) = -66
-66 is not greater than 0.
Inequality does not hold good in this range.

Range 2: x > 4
Check whether x = 5 will satisfy the inequality.
(5 - 4)(5 + 21) = 26
26 is greater than 0.
Inequality holds good in this range.

The smallest positive integer greater than 4 is 5.

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