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If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be

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parkhydel
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If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   27 Apr 2020, 14:25
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If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


PS29580.02
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E
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BrentGMATPrepNow
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If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   Updated on: 07 Oct 2020, 08:04
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parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


Key idea: \(x^2\) is always greater than or equal to 0
This means \(-x^2\) is always less than or equal to 0
And this means \(-x^2 - 3\) is always NEGATIVE
In other words, the denominator of \(y\) is most definitely NEGATIVE

From here, there are two possible cases:
case i: If the numerator of \(y\) is NEGATIVE, then y is POSITIVE
case ii: If the numerator of \(y\) is POSITIVE, then y is NEGATIVE

Since we're trying to MAXIMIZE the value of y, we need the denominator to be negative (which would make y POSITIVE)
However, the numerator, |3x - 5|, will always be greater than or equal to 0.
Since we can't make the numerator NEGATIVE, the only way to maximize the value of y is make the numerator ZERO
In other words: \(|3x - 5|=0\)
Which means \(3x -5=0\)
So \(3x =5\)
Ergo \(x = \frac{5}{3} \)

Answer: E

Cheers,
Brent

Originally posted by BrentGMATPrepNow on 27 Apr 2020, 21:23.
Last edited by BrentGMATPrepNow on 07 Oct 2020, 08:04, edited 1 time in total.
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If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   27 Apr 2020, 14:38
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|3x-5| is always non-negative and -(x^2+3) is alway negative

\( \frac{|3x - 5|}{(-x^2 - 3)} ≤ 0\)

Hence maximum value of \(\frac{|3x - 5|}{-x^2 - 3}\) is 0.

\(|3x-5| = 0\)

\(x= \frac{5}{3}\)




parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


PS29580.02
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   17 Jan 2021, 07:27
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   27 Apr 2020, 21:38
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parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


PS29580.02


We have \(y = \frac{|3x - 5|}{-x^2 - 3}\)

The denominator, \(-x^2 - 3\) MUST be Negative

The Numerator, \(|3x - 5|\) can NOT be Negative

i.e. \(y = \frac{|3x - 5|}{-x^2 - 3}\) MUST be Non-Positive



i.e. Maximum Value of a non-Positive function can be ZERO when Numerator is Zero[


i.e. 3x-5 = 0
i.e. y is maximum at \(x = 5/3\) and \(y_{max}=0\)

Answer: Option E
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   10 May 2020, 06:40
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parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


PS29580.02

Confused of signs ? just substitute X= options

arrive at Solution which gives greatest takes 1 to 2 mins. But helps if u are confused

E.5/3 gives greatest value when substituted

HOPE this helps.
THANKS :thumbsup:
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   23 May 2020, 14:35
parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


PS29580.02


Segment the numerator and denominator to look for extreme combination: |3x-5|>=0 and \(-x^2-3<=-3\), and hence y<=0.

Thus the greatest value of y is 0 --> 3x-5 = 0 --> x = \(5/3\)
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   25 Sep 2020, 10:49
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parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


PS29580.02

\(y = \frac{+ numerator }{ - denominator}\)
Hence y will always be negative except when the numerator is zero which is the largest value y can take.

\(\implies\) |3x-5| = 0
\(\implies x = \frac{5}{3}\)

Answer E.
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   07 Oct 2020, 07:55
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BrentGMATPrepNow wrote:
parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


Key idea: \(x^2\) is always greater than or equal to 0
This means \(-x^2\) is always less than or equal to 0
And this means \(-x^2 - 3\) is always NEGATIVE
In other words, the denominator of \(y\) is most definitely NEGATIVE

From here, there are two possible cases:
case i: If the numerator of \(y\) is NEGATIVE, then y is POSITIVE
case ii: If the numerator of \(y\) is POSITIVE, then y is NEGATIVE

Since we're trying to MAXIMIZE the value of y, we need the denominator to be negative (which would make y POSITIVE)
However, the denominator, |3x - 5|, will always be greater than or equal to 0.
Since we can't make the numerator NEGATIVE, the only way to maximize the value of y is make the numerator ZERO
In other words: \(|3x - 5|=0\)
Which means \(3x -5=0\)
So \(3x =5\)
Ergo \(x = \frac{5}{3} \)

Answer: E

Cheers,
Brent


Hi BrentGMATPrepNow Brent, the explanation is nice :) i just have a few questions:

When i saw :dazed |3x - 5| i thought i should consider two case

3x-5 or -3x+5 so I kinda confused...

3x=5 or -3x=-5

x = 5/3 or x = 5/3 (negative/negative =positive) both options give the same results :? :) thats strange to me Why ? :lol:

General question: if the absolute value by definition is distance from 0 and the distance is always positive or zero. why do we consider two cases positive and negative :?

p.s. i think you meant \(-x^2 - 3\) in the highlighted part
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   07 Oct 2020, 08:04
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dave13 wrote:

Hi BrentGMATPrepNow Brent, the explanation is nice :) i just have a few questions:

When i saw :dazed |3x - 5| i thought i should consider two case

3x-5 or -3x+5 so I kinda confused...

3x=5 or -3x=-5

x = 5/3 or x = 5/3 (negative/negative =positive) both options give the same results :? :) thats strange to me Why ? :lol:

General question: if the absolute value by definition is distance from 0 and the distance is always positive or zero. why do we consider two cases positive and negative :?

p.s. i think you meant \(-x^2 - 3\) in the highlighted part


You have taken the equation |3x - 5| = 0 and concluded that EITHER 3x - 5 = 0 OR -(3x - 5) = 0
Noticed that both of these equations are the same.
If we take -(3x - 5) = 0 and multiply both sides by -1, we got the equivalent equation 3x - 5 = -0, which is identical to the equation 3x - 5 = 0

Thanks for the heads-up about \(-x^2 - 3\)
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If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   23 Nov 2020, 04:54
BrentGMATPrepNow wrote:
parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


Key idea: \(x^2\) is always greater than or equal to 0
This means \(-x^2\) is always less than or equal to 0
And this means \(-x^2 - 3\) is always NEGATIVE
In other words, the denominator of \(y\) is most definitely NEGATIVE

From here, there are two possible cases:
case i: If the numerator of \(y\) is NEGATIVE, then y is POSITIVE
case ii: If the numerator of \(y\) is POSITIVE, then y is NEGATIVE

Since we're trying to MAXIMIZE the value of y, we need the denominator to be negative (which would make y POSITIVE)
However, the numerator, |3x - 5|, will always be greater than or equal to 0.
Since we can't make the numerator NEGATIVE, the only way to maximize the value of y is make the numerator ZERO
In other words: \(|3x - 5|=0\)
Which means \(3x -5=0\)
So \(3x =5\)
Ergo \(x = \frac{5}{3} \)

Answer: E

Cheers,
Brent


BrentGMATPrepNow this time i picked A :lol: because i thought since Absolute value is always positive i can pick -5 so and in denominator since it it is --
-x^2 after plugging in -5, the sign will change into positive - *-5^2 = 25

So
\(y = \frac{|3 (-5) - 5|}{- -5^2 - 3}\)

= 10/22 greater than 0

why its wrong with it? :)
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   23 Nov 2020, 06:47
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dave13 wrote:
BrentGMATPrepNow this time i picked A :lol: because i thought since Absolute value is always positive i can pick -5 so and in denominator since it it is --
-x^2 after plugging in -5, the sign will change into positive - *-5^2 = 25

So
\(y = \frac{|3 (-5) - 5|}{- -5^2 - 3}\)

= 10/22 greater than 0

why its wrong with it? :)


Think of -x² as -(x²)
So, -x² - 3 = -(x²) - 3
So, when we plug in x = -5, we get: -(x²) - 3 = -((-5)²) - 3 = -(25) - 3 = -28

Does that help?
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   03 Sep 2021, 22:19
IMO - E
I did it by subsituting the given value of X in option and found the largest value for Y is obtained by option E (5/3)
also realized, is denominator is negative and numerator is positive,
to obain highest value Y when the numerator is zero
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   04 Sep 2021, 05:02
Denominator is always negative -x^2 - 3

Max value can be 0

x = 5/3

E
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   13 Sep 2022, 09:04
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BrentGMATPrepNow wrote:
parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


Key idea: \(x^2\) is always greater than or equal to 0
This means \(-x^2\) is always less than or equal to 0
And this means \(-x^2 - 3\) is always NEGATIVE
In other words, the denominator of \(y\) is most definitely NEGATIVE

From here, there are two possible cases:
case i: If the numerator of \(y\) is NEGATIVE, then y is POSITIVE
case ii: If the numerator of \(y\) is POSITIVE, then y is NEGATIVE

Since we're trying to MAXIMIZE the value of y, we need the denominator to be negative (which would make y POSITIVE)
However, the numerator, |3x - 5|, will always be greater than or equal to 0.
Since we can't make the numerator NEGATIVE, the only way to maximize the value of y is make the numerator ZERO
In other words: \(|3x - 5|=0\)
Which means \(3x -5=0\)
So \(3x =5\)
Ergo \(x = \frac{5}{3} \)

Answer: E

Cheers,
Brent


Hi Brent,

Thank you for the great explanation. I understood the concept based on your walk-through.

Although, I did catch one issue with your one statement and please correct me if I am wrong.

BrentGMATPrepNow wrote:
Since we're trying to MAXIMIZE the value of y, we need the denominator to be negative (which would make y POSITIVE)


Shouldn't the statement above read "Numerator" instead of "Denominator"?

Regards,
Sud
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   14 Sep 2022, 19:29
BrentGMATPrepNow wrote:
parkhydel wrote:
If \(y = \frac{|3x - 5|}{-x^2 - 3}\) for what value of x will the value of y be greatest?

A. -5
B. -3/5
C. 0
D. 3/5
E. 5/3


Key idea: \(x^2\) is always greater than or equal to 0
This means \(-x^2\) is always less than or equal to 0
And this means \(-x^2 - 3\) is always NEGATIVE
In other words, the denominator of \(y\) is most definitely NEGATIVE

From here, there are two possible cases:
case i: If the numerator of \(y\) is NEGATIVE, then y is POSITIVE
case ii: If the numerator of \(y\) is POSITIVE, then y is NEGATIVE

Since we're trying to MAXIMIZE the value of y, we need the denominator to be negative (which would make y POSITIVE)
However, the numerator, |3x - 5|, will always be greater than or equal to 0.
Since we can't make the numerator NEGATIVE, the only way to maximize the value of y is make the numerator ZERO
In other words: \(|3x - 5|=0\)
Which means \(3x -5=0\)
So \(3x =5\)
Ergo \(x = \frac{5}{3} \)

Answer: E

Cheers,
Brent


Hi Brent,

I have a question regarding the modulus of the numerator.....since this is an absolute value function, the sign of the numerator will depend on the range of X. In this case, if X<5/3 the function should be written as -(3X-5), which makes the numerator negative for all range of values of X<5/3. So there can be values greater than 0 now.
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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink] New post   09 Dec 2023, 03:58
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\(y = \frac{|3x - 5|}{-x^2 - 3}\)

= \(\frac{|3x - 5|}{ -(x^2 + 3)}\)
= - \(\frac{|3x - 5|}{x^2 + 3}\)

Now, both |3x - 5| and \(x^2 + 3\) are non-negative values and we have a negative sign outside
=> Value can be maximum only when the numerator is either zero or is equal to denominator so that we get a small negative value overall

|3x - 5|=0
=> x = \(\frac{5}{3}\)

So, y = 0 when \(\frac{5}{3}\)

So, Answer will be E
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

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Re: If y = |3x - 5|/(-x^2 - 3) for what value of x will the value of y be [#permalink]
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