Inequality and absolute value questions from my collection

Hi, Logan. Good way of thinking. Though I think that your solution is not correct.

Consider this:

We have |x+2|=|y+2|

(1) xy<0, hence x and y have opposite signs. Let's take x negative and y positive (obviously it doesn't matter which one we pick as equation is symmetric).

If y is positive RHS |y+2| will be positive as well and we can expend it as |y+2|=y+2.

Now for |x+2| we can have to cases:
A. -2<x<0 --> |x+2|=x+2=y+2 --> x=y. BUT: it's not valid solution as x and y have opposite signs and they can not be equal to each other.

B. x<=-2 --> |x+2|=-x-2=y+2 --> x+y=-4. Already clear and sufficient.

If we go one step further to see for which x and y is this solution is valid, we'll get:
As x+y=-4, x<=-2 and y>0 --> x must be <-4. If you substitute values of x<-4 you'll receive the values of y>0 and their sum will always be -4.

The same approach works for (2) as well.

Hope it's clear.

Wonderful question, I solved it up to an extent but in reasoning I messed up.
Question for the quoted part which talks about the scenarios to reducing to two scenarios. I can think and verfy also though the validity of it, but want to drill further and understand how could we generalise this scenario if we could ?

Like what if |x+3|=|X-2| or |x+3|=|X-2| + |X+4| etc , just crude examples I have put , How could we generalise such scenarios ? instead of imagining six scenarios or more .

How do you conclude A here can't make out or I am tired.; anyways, it is mental marathon .. wonderful questions Bunuel. Any more link for inequality as I need some more practise .

We have \(-s<=r<=s\) --> \(-s<=s\). Now if \(s\) in negative, let's say \(-2\), then we would have \(-(-2)<=-2\) --> \(2<=-2\), which is not right. Hence \(s\) can not be negative. But \(s\) can be zero --> \(0<=0\), true.

We can count the # of scenarios for the above examples as well, but as in these examples there is only one variable we can directly solve them using the ranges, so no need to use the approach we used in solving the original question.

|x+3|=|x-2|, two check points -3 and 2, thus we'll have three ranges in which we should expand the absolute values:

A. x<-3 --> -(x+3)=-(x-2) --> -3=2, which is not true, hence in this range we have no solution;
B. -3<=x<=2 --> x+3=-(x-2) --> x=-1/2, which IS in the range we are expanding so it's a valid solution;
C. x>2 --> x+3=x+2 --> 3=2, which is not true, hence in this range we have no solution.

The equation |x+3|=|x-2| has only one solution x=-1/2. If you look at the scenarios A,B and C you'll see that |x+3|=|x-2| can take also only TWO forms x+3=x+2 or x+3=-(x-2), but the ranges are important here and we should consider these forms in their respective ranges.

Hope it's clear.

I think, when statements are given we have to answer whether based on statements given question can be answered, we should not question the validity of statements or a statements derived form the statements.
after deriving from statement 1 and 2 we deduce that x^2+y^2 =5a, this is enough to show that the question asked x^2+y^2>4a is satisfied. we are not supposed to validate whether statement x^2+y^2 =5a holds.
if u try to put x,y as 0 then just ask your self what question is trying to ask?
is 0>0.
so answer has to be C

Is \(x^2 + y^2 > 4a\)?
(1) \((x+y)^2 = 9a\)
(2) \((x-y)^2 = a\)

Consider two cases:

1. \(x=2\), \(y=1\) and \(a=1\). These values satisfy both statements and the answer to the question "is \(x^2+y^2>4a\) true" is YES, as \(5>4\) is true. (Note here that these values naturally satisfy \(x^2+y^2=5a\) too)

2. \(x=0\), \(y=0\) and \(a=0\). These values ALSO satisfy both statements and the answer to the question "is \(x^2+y^2>4a\) true" is NO, as LHS is \(0\), RHS is also \(0\) and \(0>0\) is NOT TRUE. (Note here that these values naturally satisfy \(x^2+y^2=5a\) too)

Two different answers. Not sufficient.

Answer: E.

So question is not asking whether \(0>0\). Question is asking whether "\(x^2+y^2>4a\) true". We got that for smoe values YES and for some values NO.

Hope it's clear.

You can't write it this way. -s<=r<=s means that either r lies between s & -s or r is equal to -s or r can be equal to s. But this certainly doesn't mean that you can equate -s<=s.

If I say that -4<=x<=4. This doesn't mean that I can write -4<=4.

But in any case the answer you've arrived at is correct. We can't derive anything out of the first statement as it says that -s<=r<=s. r can be anything -s or s or between -s & s.

Second says that lrl>=s ---> r>=s or r<=-s. This again is not giving any specific value.

If we combine the two we get that either r=s or r=-s. This is closer but still ambiguous. So we don't know whether r=s. Therefore, answer is E.

First of all: you are excluding the possibility when -s=r=s=0.

-s<=r<=s, so -s can be equal to s, when s=0. In all other cases -s will be less than s, so there is nothing wrong in writing this as -s<=s.

Second of all: we CAN conclude one more thing from this statement: s is either positive or zero.

Here is what I wrote in my solution from page 3:

(1) -s<=r<=s, we can conclude two things from this statement:
A. s is either positive or zero, as -s<=s;
B. r is in the range (-s,s) inclusive, meaning that r can be -s as well as s.
But we don't know whether r=s or not. Not sufficient.

(2) |r|>=s, clearly insufficient.

(1)+(2) -s<=r<=s, s is not negative, |r|>=s --> r>=s or r<=-s. This doesn't imply that r=s, from this r can be -s as well.
Consider: s=5, r=5 --> -5<=5<=5 |5|>=5
s=5, r=-5 --> -5<=-5<=5 |-5|>=5
Both statements are true with these values. Hence insufficient.

Answer: E.

Hope it's clear.

How do u get these boundary values. Looking at the equation, I solved it as x(x-2)<=0.... which gives x<=0 or x<=2... I know this isnt correct but can u let me know how u got... x>=0 & x <=2... Thanks

"Boundary values" are the roots of the equation \(x(x-2)=0\) --> \(x=0\) and \(x=2\).

I use "parabola approach" for this kind of quadratic inequalities. Basically as \(x(x-2)=0\) is an equation of upward parabola, \(x(x-2)\leq{0}\) (\(x^2-2x\leq{0}\)) is the part of the parabola, which does not lie above the X-axis. Parabola \(x(x-2)\leq{0}\) does not lie above the X-axis for the values of x in the range \(0\leq{x}\leq{2}\). If you plug the values from this range you'll see that inequality, \(x(x-2)\leq{0}\), holds true for them.

Consider inequality \((x-3)(x-7)>{0}\) (or \(x^2-10x+21>0\)). This is also upward parabola (as the coefficient of x^2 is positive), with boundary values \(x=3\) and \(x=7\). We want to determine for which value of x, this parabola is above (as we have > sign) the X-axis. The answer would be \(x<3\) and \(x>7\).

OR:

\((7+x)(3-x)>0\) (or \(-x^2-4x+21>0\)), this would be downward parabola, with boundary values \(x=-7\) and \(x=3\). This parabola lies above the X-axis for \(-7<x<3\). We can also write this as \(x^2+4x-21<0\) (or \((7+x)(x-3)<0\)), this would be upward parabola, with the same boundary values: \(x=-7\) and \(x=3\). This parabola lies below the X-axis for \(-7<x<3\), the same range as it should be.

Hope it helps.

Bunuel,

Thanks so much for all these questions and explanations!

I'm having troubles to understand the quoted question. I don't understand your explanation for statement 1, it seems like a great way to solve this problem quickly but I really just don't understand how you get there.

From this 3|x^2 -4| = y - 2 how can you conclude that y must be >=2 ?

Thx for the help!

\(3|x^2 -4| = y - 2\) --> Left hand side (LHS) is an expression with absolute value, absolute value is never negative (\(|expression|\geq{0}\)) --> \(3|x^2-4|\geq{0}\). So as \(LHS\geq{0}\), then RHS also must be more than zero (\(RHS\geq{0}\)) --> \(y-2\geq{0}\) --> \(y\geq{2}\).

Hope it's clear.

I am a little confused on this one . Can the answer be E??

From A:
2x-2y=1
=> x-y= 0.5 INSF

From B
x/y > 1
=> x > y INSF

From A & B
x-y =0.5 and x > y

If x = -0.5 and y = -1 then
x > y
and x - y = (-0.5) - (-1) = -0.5 + 1 = 0.5
Hence both x and y can be negative

If x= 1 and y = 0.5 then
x > y
and x- y = 1 -0.5 = 0.5
Hence both x and y can be positive

Ans = E ??

Problem with your solution is that the red part is not correct.

\(\frac{x}{y}>1\) does not mean that \(x>y\). If both x and y are positive, then \(x>y\), BUT if both are negative, then \(x<y\).

From (2) \(\frac{x}{y}>1\), we can only deduce that x and y have the same sigh (either both positive or both negative).

When we consider two statement together:

From (1): \(2x-2y=1\) --> \(x=y+\frac{1}{2}\)

From (2): \(\frac{x}{y}>1\) --> \(\frac{x}{y}-1>0\) --> \(\frac{x-y}{y}>0\) --> substitute \(x\) from (1) --> \(\frac{y+\frac{1}{2}-y}{y}>0\)--> \(\frac{1}{2y}>0\) (we can drop 2 as it won't affect anything here and write as I wrote \(\frac{1}{y}>0\), but basically it's the same) --> \(\frac{1}{2y}>0\) means \(y\) is positive, and from (2) we know that if y is positive x must also be positive.

OR: as \(y\) is positive and as from (1) \(x=y+\frac{1}{2}\), \(x=positive+\frac{1}{2}=positive\), hence \(x\) is positive too.

Answer: C.

Hope it's clear.

Ive got C for this question..

when both together yields x^2 + y^2 = 5a
why it is E?

Also I dont understand the explanation of below,
St. (1) and (2) together : x^2 + y^2 = 5a
When either x or y is not 0, question stem holds true.
When x and y are both 0, question stem is false.

Can somebody clarify how to solve this please...

OA' s and solutions for all the problems are given in my posts on pages 2 and 3.

OA for this question is E.

When we consider statement together we'll have: \(x^2+y^2=5a\). Now, if \(x\), \(y\) and \(a\) are different from zero (for example: \(x=3\), \(y=4\) and \(a=5\)) then \(x^2+y^2=5a>4a\) and the answer to the question is YES, but if \(x=y=a=0\), then \(x^2+y^2=5a=4a=0\) and the answer to the question is NO. Two different answers, thus not sufficient.

Hope it's clear.

Hi Bunuel,
I forgot to thank you for these great questions and solutions. You are the BEST!!!!

If y is an integer and y = |x| + x, is y = 0?
(1) x < 0
(2) y < 1

(2) y<1, as we concluded y is never negative, and we are given that y is an integer, hence . Sufficient.

In the above problem, it is very obvious that Y is not equal to 0 from the 2nd statement.
Do you think GMAT questions will have this much obvious statements?