For what range of values of 'x' will the inequality 15x - 2/x 1?

Question

For what range of values of 'x' will the inequality 15x - 2/x > 1? A. x > 0.4 B. x < 1/3 C. -1/3 < x < 0.4, x > 15/2 D. -1/3 < x < 0, x > 2/5 E. x < -1/3 and x > 2/5 D

Bunuel · Accepted Answer

anshumishra considered 2 cases for x in order to multiply both parts of the inequality by x and simplify it by doing so. One could also do as follows: 15x-\frac{2}{x}>1 --> \frac{15x^2-x-2}{x}>0 --> \frac{(x+\frac{1}{3})(x-\frac{2}{5})}{x}>0 : Both denominator and nominator positive --> x>\frac{2}{5} ; Both denominator and nominator negative --> -\frac{1}{3}\frac{2}{5} . Answer: D.

anshumishra · Answer

15x - 2/x > 1

Case1 : x > 0
Multiply both sides by "x"
=> 15x^2 - 2 > x
=> 15x^2 - x - 2 > 0 (solve the quadratic eqn for 15x^2 - x- 2 = 0 , it gives you : x = -1/3 or 2/5)
Since x > 0 , so x > 2/5

Case 2 : x < 0
Multiply both sides by "x"
=> 15x^2 - 2 < x
=> 15x^2 - x - 2 < 0
Since, x < 0 so, the range of permissible value is : -1/3 < x < 0.

devashish · Answer

I do not understand why 2 cases are considered in the above explanation ?
Only if we had |x| should the 2 cases be considered, am I right ?

anshumishra · Answer

Whenever you multiply an inequality by X , you should check for the -ve and +ve values of X, as it inverts the sign.

devashish · Answer

Thanks for the lightning quick reply, I was not aware of this.

devashish · Answer

Bunuel you are awesome

anshumishra · Answer

Nice alternate solution.

Please note :

15x^2-x-2 -> x^2 - x/15 - 2/15 -> x^2 + x/3 - 2x/5 - 2/15 -> (x+1/3)(x-2/5)

This works well, however I would suggest to rely on quadratic equation formula:  -b(+-)\sqrt{D}/2a , as you don't need to think, how to factorize an equation?

Try yourself to factorize the equation above vs using the formula (Unfortunately, if the equation doesn't has a fractional value (which I expect rarely could be the case in GMAT) , for example when D in above formula is irrational, then you might be trying to factorize the fraction forever.)

PiyushK · Answer

What I have learnt, if x is in denom, take LCM after bringing all element at one side. If we have more than 2 roots, then we can use following method to know the range. Solution.jpg

rrsnathan · Answer

hi,

can any one explain this with graphical explanation??

Thanks in advance,
Rrsnathan.

TooLong150 · Answer

In Case 1 where x > 0, I agree. I calculated x > (-1/3) and x > 2/5, and because x > 0, it must be x > 2/5. In Case 2 where x < 0, I don't understand why you say that the range must be x > -1/3 and x < 0, since the values that satisfy the inequality "15x^2 - x - 2 < 0" show that x < -1/3 and x < 2/5.

siriusblack1106 · Answer

I have the same doubt. Shouldn't the answer be D?

Bunuel · Answer

____________
The answer IS D.

m3equals333 · Answer

After you determine boundary points, you still need to test the potential +ve/-ve ranges per the inequality

Enael · Answer

Hi Bunuel, I thought that we shouldn't multiply an inequality with a variable whose range of value could be +ive and/or -ive.

Bunuel · Answer

I didn't multiply.

15x-\frac{2}{x}>1  --> subtract 1:  15x-\frac{2}{x}-1>0  --> put in common denominator:  \frac{15x^2-x-2}{x}>0  --> factorize:  \frac{(x+\frac{1}{3})(x-\frac{2}{5})}{x}>0 .

Hope it's clear.

GSBae · Answer

A graphical approach supplements this solution quite well: The roots are the blue dots, the yellow line separates the two cases, and the red lines are where the inequality holds true. We assume two cases: x>0 and x<0. For the first case, x>0, we want to see where 15x^2 -x-2 >0. Solving for the roots using the quadratic formula, we find roots when x = 2/5 and x = -1/3. Thus, for x>0, 15x^2 -x-2 >0 when x>2/5. For the second case, x<0, we want to see where 15x^2 -x-2 <0. Again, using the roots, we know that the 15x^2 -x-2 < 0 when x > -1/3. We also assumed x<0 so we must include that in our solution. Thus we have x>2/5 and -1/3

GGMAT760 · Answer

Hi 
Sorry for asking basic question. But do you have detail link to solve complicated quadratic equations like this: 15x2-x-2 ? 
I don't know how to solve this????

Bunuel · Answer

Theory on Inequalities:
Solving Quadratic Inequalities - Graphic Approach: solving-quadratic-inequalities-graphic-approach-170528.html
Inequality tips: tips-and-hints-for-specific-quant-topics-with-examples-172096.html#p1379270

inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html

All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184
All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189

700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html

ronr34 · Answer

Can you explain why before and after the "0" you change the sign?
What was the equation that got you to draw this?

For what range of values of 'x' will the inequality 15x - 2/x > 1?

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