inequality, modulus and 2 variables

Question

How can we express y in terms of x. More precisely, how to form different cases to solve the below inequality |10y - 4| < |10x + 6| Regards, Vishal

theCodeToGMAT · Answer

|10y - 4| < |10x + 6| ==> 10y - 4 < 10x + 6 ==> 10y < 10x + 10 ==> y -(10y - 4) < 10x + 6 ==> 4 - 10y < 10x + 6 ==> 10y + 10x > -2 ==> 10y > -2 -10x

Mathsbuddy · Answer

How can we express y in terms of x. More precisely, how to form different cases to solve the below inequality

|10y - 4| < |10x + 6|

As both sides are positive, we can square both sides without conflict:

(10y - 4)^2 < (10x + 6)^2

(10y - 4)^2 - (10x + 6)^2 < 0

100y^2 -80y + 16 - 100x^2 - 120x - 36 < 0

100y^2 -80y - 100x^2 - 120x - 20 < 0

Would that start our journey in the wrong direction?

Haldiram Bhujiawala · Answer

vishalpathak wrote:How can we express y in terms of x. More precisely, how to form different cases to solve the below inequality

|10y - 4| < |10x + 6|

Please check the Manhattan guides for ways to form different cases to solve the inequality. The guide provides different scenarios for evaluating different cases. It's very comprehensive however a bit lengthy. Alternatively, since both sides of inequality have modulus value expresions, the values of the expression will be positive. So we can safely consider only the positive value of each expresion and solve accordingly. Rest part of the question can be solved as shown by Mathsbuddy. I hope there won't be any difficulty solving mathsbuddy's way.

vishalpathak · Answer

Haldiram Bhujiawala wrote:vishalpathak wrote:How can we express y in terms of x. More precisely, how to form different cases to solve the below inequality |10y - 4| < |10x + 6| Please check the Manhattan guides for ways to form different cases to solve the inequality. The guide provides different scenarios for evaluating different cases. It's very comprehensive however a bit lengthy. Alternatively, since both sides of inequality have modulus value expresions, the values of the expression will be positive. So we can safely consider only the positive value of each expresion and solve accordingly. Rest part of the question can be solved as shown by Mathsbuddy. I hope there won't be any difficulty solving mathsbuddy's way. When we square both sides of an equation, we add aditional root to the equation. I am not sure whether squaring is the right method. However, I do not know the correct method myself Regards, Vishal

vishalpathak · Answer

the following comes to my mind for this problem

when x < -6/10, 10x + 6 is -ve
Hence

0 < 10y-4 < -(10x + 6) -- (y has to be > 4/10 for this eqn to hold true)

when x > -6/10 10x + 6 is +ve
Hence

0 < 10y-4 < (10x + 6) -- (y has to be > 4/10 for this eqn to hold true)

This can help us find a range of y in terms of x

Similarly we can have 2 more eqns when y < 4/10

These 4 eqns can help us find the required ranges.

I am not confident about the solution though

GMATSUCKER · Answer

vishalpathak wrote:
When we square both sides of an equation, we add aditional root to the equation. I am not sure whether squaring is the right method. However, I do not know the correct method myself

When you find the roots of the Quadratics you should always plug back the roots in the original inequation to check whether they satisfy the original inequation. If they satisfy then generally they are the solution to the inequation.

As a general rule an expression represents a quantity or an idea. So if you square both sides of the inequation it does not affect the inequality. And squaring is the most safest way to ensure that the quantity on both sides are actually positive so as to be consistent with the modulus. I think Mathsbuddy will be able to shed more light on the approach.

Last edited by GMATSUCKER on Sat Nov 23, 2013 1:22 am, edited 1 time in total.

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Rich.C@EMPOWERgmat.com · Answer

Hi All,

The GMAT Quant section is not a "math test"; it's a test of your critical thinking skills and math is the subject through which those skill tests are run. While you will end up doing plenty of math in that section, it's important to know what the concepts are "in context" of the question that is asked AND what appears in the 5 answer choices. In many cases, "doing math" is not the most efficient way to get to the correct answer.

All that having been said, what is the context for the attached math inequality? What were you asked? What is in the 5 answer choices?

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

vishalpathak · Answer

Rich.C@EMPOWERgmat.com wrote:Hi All, The GMAT Quant section is not a "math test"; it's a test of your critical thinking skills and math is the subject through which those skill tests are run. While you will end up doing plenty of math in that section, it's important to know what the concepts are "in context" of the question that is asked AND what appears in the 5 answer choices. In many cases, "doing math" is not the most efficient way to get to the correct answer. All that having been said, what is the context for the attached math inequality? What were you asked? What is in the 5 answer choices? GMAT assassins aren't born, they're made, Rich Hi Rich, My attempt is to formulate a strategy to solve such problems. I understand your point about looking at the options and not doing maths but I was wondering if I have to do maths then what would be the process for a question like this I do not have a question that involves 2 variables, modulus and inequality but if there was one then what would be the process to solve it? Regards, Vishal

Rich.C@EMPOWERgmat.com · Answer

Hi vishalpathak,

In Absolute Value/Modulus questions, I usually end up TESTing Values - it's fast, easy and works on most of these types of questions.

Based on what you've claimed (that you have seen NO questions that fit these parameters), then I have to ask WHY you're focused on a math concept that you haven't seen, and likely won't see, on the GMAT? It seems like a bad use of your time.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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