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# The dark purple region on the number line above is shown in its entire

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Magoosh GMAT Instructor
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
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mikemcgarry wrote:
Attachment:

The dark purple region on the number line above is shown in its entirety. This region is delineated by which of the following inequalities?

(A) 10 < |x + 10| < 80

(B) 10 < |x – 100| < 80

(C) |x – 20| < 70

(D) |x – 20| < | x – 90|

(E) |x – 55| < 35

Another approach is to check whether possible solutions for each answer choice match up with the solution shown on the number line.

(A) 10 < |x + 10| < 80
One possible solution to this inequality is x = 5
However, the number line does NOT include x = 5 as a solution
ELIMINATE A

(B) 10 < |x – 100| < 80
One possible solution to this inequality is x = 130
However, the number line does NOT include x = 130 as a solution
ELIMINATE B

(C) |x – 20| < 70
One possible solution to this inequality is x = 0
However, the number line does NOT include x = 0 as a solution
ELIMINATE C

(D) |x – 20| < | x – 90|
One possible solution to this inequality is x = 10
However, the number line does NOT include x = 10 as a solution
ELIMINATE D

By the process of elimination, the correct answer is E

Cheers,
Brent
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
The range we are looking for is 90>=x>=20. Both B and E offer that range. Why is B incorrect? Can an expert please advise?

Thank you
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
mikemcgarry wrote:
OreoShake wrote:
The range we are looking for is 90>=x>=20. Both B and E offer that range. Why is B incorrect? Can an expert please advise?

Thank you

Dear OreoShake,

I'm happy to respond.

We are looking for an inequality that expresses exactly that range--an inequality that includes every single point on that purple line and excludes every single point not on the purpose line. (E) does this perfectly.

The problem with (B) is that it certainly includes everything on the purple line, but it also includes a bunch of other points that are not on the line. For example, the points x = 100 and x = 150 both satisfy the inequality in (B), but these are points not included in the purple region.

Under-inclusion and over-inclusion are opposites, but they are both problems. We are looking for an exact fit.

Does this make sense?
Mike

Thank you Mike, I understand your explanation. Would there be any cues in gmat question stem that indicate that we need an exact range? Or are we supposed to infer that for questions that ask you to find a matching range?
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
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OreoShake wrote:
Thank you Mike, I understand your explanation. Would there be any cues in gmat question stem that indicate that we need an exact range? Or are we supposed to infer that for questions that ask you to find a matching range?

Dear OreoShake,

I'm happy to respond.

Mathematics is always about precision 100% of the time. As a general rule, if you are unclear how specific the question wants you to be, an one answer is an exact fit, and the other is not, the exact fit is the right answer.

I don't know if you are familiar with the verb "delineate," a somewhat difficult vocabulary word to appear in a math prompt?
This region is delineated by which of the following inequalities?
If you can't give an exact definition of this word, then that's an impediment to understanding the question.

to delineate - to show the lines around something; hence, to indicate the exact boundaries of something

If you have questions about a GMAT math problem, and you cannot give the exact definition of a word that appears in the prompt, then as part of understanding the problem, you need to go on the web and figure out the exact meaning of that word. Everything in math is precise, including the words used in a problem. You cannot afford to overlook or skip the meaning of a single syllable in a math problem.

Does all this make sense?
Mike
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
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Perhaps the option E could be looked at like this :
1. For +ve values of x
x-55 < 35, i.e. x<90
2. For -ve values of x
-x + 55 < 35 -->
Adjusting the -ve sign results in x - 55 > -35
i.e. x > 20
So, 20 < x <90
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
manishtank1988 wrote:
Step one: find the midpoint of the region. The midpoint, halfway between and 20 and 90, is 55. In other words, 20 and 90 have the same distance from 55, a distance of 35. These endpoints are not included, but the region includes all the points that have a distance from from x = 55 that is less than 35. Translating that into math, we get the following:
|x – 55| < 35
Source: magoos url provided!!!

What a new perspective to absolute value questions !!!!!
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
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mikemcgarry wrote:
Attachment:

The dark purple region on the number line above is shown in its entirety. This region is delineated by which of the following inequalities?

(A) 10 < |x + 10| < 80

(B) 10 < |x – 100| < 80

(C) |x – 20| < 70

(D) |x – 20| < | x – 90|

(E) |x – 55| < 35

One option is to test each answer choice.

In cases where we must check/test each answer choice, always check the answer choices from E to A, because the correct answer is typically closer to the bottom than to the top. For more on this strategy, see my article: https://www.gmatprepnow.com/articles/han ... -questions

E) |x – 55| < 35
We can apply the rule that says:If |something| < k, then –k < something < k
So, |x – 55| < 35 becomes -35 < x – 55 < 35
Add 55 to all 3 parts to get: 20 < x < 90
Perfect!

Cheers,
Brent
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The dark purple region on the number line above is shown in its entire [#permalink]
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mikemcgarry wrote:
OreoShake wrote:
The range we are looking for is 90>=x>=20. Both B and E offer that range. Why is B incorrect? Can an expert please advise?

Thank you

Dear OreoShake,

I'm happy to respond.

We are looking for an inequality that expresses exactly that range--an inequality that includes every single point on that purple line and excludes every single point not on the purpose line. (E) does this perfectly.

The problem with (B) is that it certainly includes everything on the purple line, but it also includes a bunch of other points that are not on the line. For example, the points x = 100 and x = 150 both satisfy the inequality in (B), but these are points not included in the purple region.

Under-inclusion and over-inclusion are opposites, but they are both problems. We are looking for an exact fit.

Does this make sense?
Mike

Hey mikemcgarry,
Thank you for such a great question. I looked at the options and intentionally started with options C and E and found option E to have exactly the range, which is asked in the question.

I just wanted to confirm a value that you have mentioned in your explanation above(highlighted in blue). You said point x = 100 and x = 150 satisfies the inequality given in option B, but I am afraid because I couldn't find how x = 100 satisfies the same.
IMO, the range of X in option B should be;
1. 20 < X < 90 (exactly what is being asked in the question) and
2. 110 < X < 180.
Both the above ranges do not include X = 100. Please let me know in case I am going wrong somewhere.

Thanks.
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
aalekhoza

Isn't the range in B:
1. 110<X<90
2. 20<X<180

?
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Re: The dark purple region on the number line above is shown in its entire [#permalink]
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aalekhoza

Isn't the range in B:
1. 110<X<90
2. 20<X<180

?

Hey Eladt, I have attached the solution. I hope it helps.
Attachments

IMG_9284.JPG [ 1.86 MiB | Viewed 8971 times ]

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Re: The dark purple region on the number line above is shown in its entire [#permalink]
I understand it now!
Thank you aalekhoza
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The dark purple region on the number line above is shown in its entire [#permalink]
Distance Interpretation of the Absolute Value Modulus:

(1st) Since the Range of Values on the Number Line is a continuous range and INCLUDES the Values at the end points, the Inequality must be of the Form:

| expression | </= K

which becomes, after Opening the Modulus:

-(K) </= (expression) </= +K

(2nd) Find the Midpoint of the Continuous Range that is Shaded purple on the Number Line

(20 + 90) / 2 = 55

Each endpoint, 90 and 20, is exactly 35 Units from the Midpoint of 55 on the Number Line

(3rd) Distance Evaluation of the Absolute Value Modulus

All of the Values within the Range shown by the Purple Line on the Number Line will satisfy the Inequality we are looking for

Therefore, from the Mid-point of 55, the Value of X must be LESS THAN or EQUAL TO 35 Units away.

This is expressed by:

| X - 55 | </= 35

The Value of X must fall within the range on the Number Line that is Less Than or Equal To a DISTANCE of 35 Units away from +55 ——in EITHER Direction

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Re: The dark purple region on the number line above is shown in its entire [#permalink]
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