GMAT Sample Questions | GMAT Inequalities Q2

This GMAT quant practice question is a problem solving question from Inequalities. Concept: Finding solution set to an algebraic inequality. Level of difficulty: Easy to moderate. GMAT 650 level inequalities sample question.

Question 2: What range of values of 'x' will satisfy the inequality 15x - \\frac{2}{x}) > 1?

1. x > 0.4
2. x < \\frac{1}{3})
3. -\\frac{1}{3}) < x < 0.4, x > \\frac{15}{2})
4. -\\frac{1}{3}) < x < 0, x > \\frac{2}{5})
5. x < -\\frac{1}{3})and x > \\frac{2}{5})

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Video Explanation

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Explanatory Answer | GMAT Algebraic Inequalities

We can rewrite the above inequality as 15x - \\frac{2}{x}) - 1 > 0
i.e., \\frac{15x^2 - 2 - x}{x}) > 0
Split the middle term of the numerator as a precursor to factorizing it: \\frac{15x^2 - 6x + 5x - 2}{x}) > 0

Factorize the quadratic expression: \\frac{(5x - 2)(3x + 1)}{x}) > 0
The above inequality will hold good if the numerator and denominator are both positive or are both negative.

Possibility 1: When (5x - 2)(3x + 1) > 0 and x > 0
Rule: (x - a)(x - b) > 0 when x does not lie between "a" and "b".
Applying the rule, the values of 'x' that will satisfy (5x - 2)(3x + 1) > 0 will not lie between -\\frac{1}{3}) and \\frac{2}{5}).
Combining the above result with the second condition that x > 0, we get x > \\frac{2}{5})

Possibility 2: When (5x - 2)(3x + 1) < 0 and x < 0.
Rule: (x - a)(x - b) < 0 when x lies between "a" and "b".
Applying the rule, the following values of 'x' will satisfy (5x - 2)(3x + 1) < 0: -\\frac{1}{3}) < x < \\frac{2}{5}).
Combining the above range of values with the second condition that x < 0, we get -\\frac{1}{3}) < x < 0.

Therefore, the range of values of x that will satisfy the inequality is: -\\frac{1}{3}) < x < 0 \\cup) x > \\frac{2}{5}).

Choice D is the correct answer.