Which of the following represents 1<x<9?

Hi, I'm happy to help with this.

Fundamentally, absolute value is about distance. That is something many folks fail to grasp.

|x| is the distance of x from the origin on the number line.

|x - 5| is the distance of x from the point 5 on the number line.

|x + 3| is the distance of x from the point -3 on the number line. (That's because x + 3 = x - (-3))

So, we want to write 1<x<9 as a distance statement.

Step #1: find the middle of that region. The midpoint is at 5. So, this is about distance from 5 ---> |x - 5|

Step #2: how far can we get from 5? Well, up to almost 9, or down to almost 1 -- that is, almost 4. Thus:

distance from 5 < 4

|x - 5| < 4

Once you understand the distance-interpretation of absolute value, you need not spend much time at all on questions like this.

Does that make sense? Please let me know if I can clarify anything here or if you have any more questions.

Here's a challenging absolute value question for practice.

https://gmat.magoosh.com/questions/964

Once you submit your answer, that question should be followed by a video explanation of the answer.

Mike
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I'd probably use the distance concept for this question the same way as Mike well explained above. Though one can also use plug-in method:

Given: 1<x<9. Now, try two numbers, one from this range and one out of this range. Correct answer choice must be true for the first number and false for the another. Let's try: x=8 and x=0. So, the correct answer choice must be true for x=8 and false for x=0.

A. |x|<3 --> false for x=8, so no need to try another number. Discard;
B. |x+5|<4 --> false for x=8, so no need to try another number. Discard;
C. |x-1|<9 --> true for x=8, but also true for x=0. Discard;
D. |-5+x|<4 --> true for x=8, and false for x=0. Correct;
E. |3+x|<5 --> false for x=8, so no need to try another number. Discard.

Only one answer choice D works.

Answer: D.

Note that for plug-in method it might happen that for some particular numbers more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only.

You can also check Absolute Value chapter of Math Book, where Walker discusses the same exact question using the distance concept: math-absolute-value-modulus-86462.html

Hope it helps.
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IMO

A. |x|<3 => -3<x<3 => no need to go further its wrong
B. |x+5|<4 => -4<x+5<4 => -9<x..... no need to go further its wrong
C. |x-1|<9 => -9<x+1<9 => -10<x..... no need to go further its wrong
D. |-5+x|<4 => -4<x-5<4 => 1<x<9 - BINGO
E. |3+x|<5 => -5<x+3<5 => -8<x..... no need to go further its wrong

Hence D

1<x<9

1) Get the center of the range
CENTER = (9 + 1)/2 = 10/2 = 5
2) Flip the sign and append the variable x
|x - 5|
3) Get the distance of the CENTER from one of the END
DISTANCE = 9 - 5 = 4
4) Decide the inequality sign. WE know that x is within the range so choose "<"

ANsweR: |x-5| < 4

The below is my approach for any modulus qtn in GMAT. Please read thru the complete explanation to understand the modulus concept.

Remember.
The meaning of |x-y| is "On the number line, the distance between X and +Y"
The meaning of |x+y| is "On the number line, the distance between X and -Y"
The meaning of |x| is "On the number line, the distance between X and 0".

On the # line, Left to 0 are all the -ve #s and right to 0 are all +ve #s

original qtn:
is x between 1 and 9 ==> .....................0..1....................9..... ==> is X in the highlighted area

A) |x| < 3 ---- wrong
on the number line, the distance of x from 0 is < 3 ==> ......-3........0........3....

B) |x+5|<4 --- wrong
on the number line, the distance of x from -5 is < 4 ==> ............-9............-5...........1...0.......3....

C)|x-1|<9 --- wrong
on the number line, the distance of x from 1 is < 9 ==> ..................-8..............................0....1.......................9...10....

D) |-5+x|<4
on the number line, the distance of x from +5 is < 4 ==> ............0..1..............5...............9 ---- This is what is asked in the question, hence, correct answer
E. |3+x|<5

Though mikemcgarry and bb explained well - I ended up solving this by solving the mods in the answer choices. And as a result took few secs extra may be.

I liked the value substitution by bb and I will try with other questions.
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FYI, bb and Bunuel, are two different people.
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Ha Ha.... You caught that..... How can i even think of crediting Maths answers to anyone else other than Bunuel I was reading some articles of bb and hence jotted down bb instead og Bunuel.... Sorry!
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By far the best response from Bunuel !! There can be only one Legend


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Such approach makes life easier and smoother while preparing for GMAT. Hopefully, I get this kind of solution for every question.
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By looking at the choices, we can make an educated guess that the answer is going to be either choice B or choice D since only 4 and 5 can sum to 9 (when added) and make 1 (when subtracted). Let’s check choice B first.

If (x + 5) is positive:

x + 5 < 4

x < -1

We can stop here since this doesn’t match the given inequality 1 < x < 9.

Now let’s check choice D.

If (-5 + x) is positive:

-5 + x < 4

x < 9

If (-5 + x) is negative:

-(-5 + x) < 4

5 - x < 4

1 < x

Thus we see that 1 < x < 9.

Answer: D
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Hi All,

We're told that 1 < X < 9. We're asked which of the absolute value inequalities properly 'matches.' This question can be solved rather easily by TESTing VALUES. We're going to test numbers against the prompt and the answers to find the correct answer:

Answer A: doesn't include certain numbers (3 - 8.99999) = eliminate
Answer B: doesn't include any of the numbers = eliminate
Answer C: Includes the above range BUT ALSO includes numbers NOT in that range (1, 0, -1, etc.) = SUSPICIOUS ANSWER
Answer D: Includes ALL numbers from that range = LOOKS GOOD
Answer E: doesn't include certain numbers (2 - 8.9999) = eliminate

Final answer:

GMAT assassins aren't born, they're made,
Rich
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|-5+x|<4 => -(-5+x)<4 or (-5+x)<4
=> -5+x>-4 or -5+x<4
=> x>1 or x<9 => 1<x<9

option D
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I understand everything until you list the steps. Why we can conclude from 1<x<9 to |x-5| ?

Clearly |-5+x|<4 is true because positive
Case: -5+x <4 —> x< 9
Negative case :
5-x < 4 —> x> 1

Posted from my mobile device

I did not doubt if it is correct, I just wonder how you get to step 1 and 2

Both positive and negative values are possible for value inside mod || which is -5+x
1) For -5+x to be positive, |-5+x| < 4 --> -5+x < 4 --> x < 9
2) For (-5+x) to be negative, |-5+x| < 4 --> -(-5+x) < 4 --> 5-x < 4 --> 5 < 4+x --> x > 1

A. |x|<3 --> Not possible as positive case : x<3, negative case: -x < 3 or x > -3
B. |x+5|<4 --> Not possible as positive case: x+5 < 4 --> x < -1
C. |x-1|<9 --> Not possible, as positive case: x-1 < 9 --> x < 10, negative case : -x+1 < 9 --> x > -8
E. |3+x|<5 --> Not possible, as positive case: 3+x < 5 --> x < 2

This might be helpful:
|x| < constant basically means (-constant) < x < (+constant) ~~[|x|<4 means -4<x<4]
|x|> constant basically means x>constant or x<-constant ~~[|x|>4 means x>4 or x<-4]

And now for how to solve such questions easily without having to go through evaluating all choices -

One look at the answer choices and all of them are of the form "|expression|<some-constant"
To bring 1<x<9 to that form, do the below steps

1 - Find the midpoint between 1 and 9, which is 5
2 - Subtract 5 from each of the three sides --> 1-5 <x-5<9-5 which becomes -4<x-5<4
3- Which basically means |x-5|<4