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

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

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

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

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

What a new perspective to absolute value questions !!!!!

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!

Answer: E

Cheers,
Brent

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.

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.

I understand it now!
Thank you aalekhoza

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


-E- is the Correct Answer

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