GMAT Avengers Study Group: Absolute Value:

by on August 21st, 2014

The absolute value of x, |x|, is the distance of x from zero on the number line. Of course, it’s always positive, because distance is away positive. To extend that further: |x – 4| is the distance of x from 4; |x – 7| is the distance of x from 7; |x + 3| is the distance of x from -3 (this is because x + 3 = x – (-3) when written as subtraction)

In other words, the absolute value (or modulus), |x|, of a real number x is x‘s numerical value without regard to its sign. It is always greater than or equal to zero.

For example: |3|=3; |-12|=12, |-1.3|=1.3;

It is always important to think a bit about how absolute values impact problems in which positive/negative properties are uncertain. And it also provides us with a bit of an overall clue for how to approach a particular problem. If the problem or any solutions have an absolute value, there’s a really high likelihood that the properties of positive/negative/zero will be coming into play.

The links of the articles that we shared on our Facebook Event Page are listed below for a quick reference. The articles talk about all the important concepts, strategies, tips pertaining to Absolute Values and thus, should make up for a happy reading :) .

Properties of Absolute Values

Knowing these properties can surely save us some time on test day:

  1. |x| ≥ 0
  2. |x|= sqrt{x^2}
  3. |0|=0
  4. |-x|=|x|
  5. |x|+|y| ≥ |x|-|y|
  6. |x - y| ≥ |x|-|y|
  7. |xy|=|x|*|y|
  8. |y|=0 means y

Problem Solving Practice

1)  If n is an integer, the greatest possible value of the expression: 12 – |32 – 7n| is

(A) -20

(B) 1

(C) 8

(D) 9

(E) 12

2) If |x|<x^2, which of the following must be true?

I. x^2>1

II. x>0

III. x<-1

 

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I and III only

3) What is sqrt{x^2*y^2}  if x < 0 and y > 0?

(A) –xy

(B) xy

(C) –|xy|

(D) |y|x

(E) No solution

4) Two prime numbers are considered consecutive if no other prime lies between them on the number line. If p_1 and p_2 are consecutive primes, with |p_1-p_2| > 2, what is the smallest possible absolute value of the coefficient of the x term in the distributed form of the expression (x-p_1)(x-p_2) ?

(A) 5

(B) 8

(C) 12

(D) 18

(E) 24

5) Which of the following represents the complete range of x over which x^3-4^x^5 < 0?

(A) 0 < |x| < 1/2

(B) |x| > 1/2

(C) -1/2 < x < 0 or 1/2 < x

(D) x < -1/2 or 0 < x < 1/2

(E) x < -1/2 or x

6) If each expression under the square root is greater than or equal to 0, what is sqrt{x^2-6x+9}+sqrt{2-x}+x-3?

(A) sqrt{2-x}

(B) 2x-6+sqrt{2-x}

(C) sqrt{2-x}+x-3

(D) 2x-6+sqrt{x-2}

(E) x+sqrt{x-2}

Data Sufficiency Practice

1) If x and y are integers and x < y, what is the value of x + y?

(1) x^y=4

(2) |x| = |y|

2) If v ≠ 0, is |w| < |v|?

(1) w/v < 1

(2) {w^2}/{v^2}< 1

3) If #x# represents the smallest even integer greater than or equal to x^2 , is #x# less than 12?

(1) |x – 3| ≤ 1

(2) |x – 1| ≤ 2

4) What is the value of |x|?

(1) x = -|x|

(2) x^2 = 4

5) Is x < 0?

(1) x^2 = 9x

(2) |x| = -x

6) Is n<0?

1) -=|-n|

2) n^2=16

Answer Key

Problem Solving

  1. D
  2. A
  3. A
  4. D
  5. C
  6. A

Data Sufficiency

  1. D - Each statement alone is sufficient to answer the question
  2. B - Statement 2 alone is sufficient but statement 2 alone is not sufficient to answer the question asked
  3. B - Statement 2 alone is sufficient but statement 2 alone is not sufficient to answer the question asked
  4. B - Statement 2 alone is sufficient but statement 2 alone is not sufficient to answer the question asked
  5. A - Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked
  6. C - Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone

Make sure you check out the event Facebook page to see all the comments that were being exchanged during the live session.

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20 comments

  • I am not that good in concepts regarding the absolute values. Can you provide me a detail explanation for the above problems.

    • Meera, could you let us know your email id so that we can send the solutions to you in email. Thanks in advance,
      Rahul !!

    • Rahul - could you also please send me the solution? Thanks in advance!!

    • Hi Rahul, 

      Would you mind emailing me the solutions? I'm getting a good number wrong and would like to make sure that I understand the concepts. Thanks. 

      Best regards,

      CG

    • I am sorry for the long delay in my response. Could you please let me know your email address so that I can send the solutions to you ?

  • For the first question,are you sure 9 is the answer?I used a graphing software to plot all the values.It seems like when f(n)=9 ,n won't be an integer.

    • Sriram - D is indeed the correct answer here.

      We are being told to calculate the greatest possible value of the expression. Now, for that to happen, the value of the absolute expression has to be zero. Now, putting the expression to zero, we get - |32 -7n| = 0. Upon solving, we will get n = 32/7 which gives value of n as 4.571. Now, we are being told that n has to be an integer. So, we will take value of n as 5 instead of 4 because that would give us the least value of the absolute expression. When we solve using the value of n as 5, we will get 12 - |-3| = 9 which is answer choice D. Hope it helps. Please do let me know in case something is still not clear. All the best to you !!

  • Rahul, how do I let you know my email ID? I'd also like to see the answers! Thanks,
    Phillip

    • Hey Phil - Could you send me a test email at sehgalrahul84@gmail.com and I will send you the solution to these questions. Hope this helps !!

  • Hi Rahul, 
    What is the explanation for P.S #5. 
    I searched for the explanation in your facebook event but looks like the Q was not posted. Any help is much appreciated. TIA

  • Rahul, could you please email me the detailed explanation to these questions. Please check your email, I have mailed you my id.
    Thanks in advance.

  • I believe that the answers posted for the question numbers 5 and 6 are wrong.

    for Q 5. if one substitutes 1 or -1 to the value x^3-4^x^5...the answers are 0..ie x<2...thus the roots of the first expression is a negative term as x-3 <0..but that also means that the last term in the question i.e x-3 is also less than <0. thus negative and negative should add up and not cancel each other.thus the answer to the Q6 is option B and not A.

  • Corrections to my last comment.....i believe that the answers posted in respect of question numbers 5 and 6 in PS section are wrong.

    for Q 5. if one substitutes 1 or -1 to the value x^3-4^x^5...the answers are less than 0..thus option choice c does not hold good at all

    next for Q6. the values under the roots are positive, thus 2-x is positive..i.e x<2...thus the roots of the first expression (x-3) is a negative term as x<2..but that also means that the last term in the question i.e x-3 is also less than <0. thus negative and negative should add up and not cancel each other.thus the answer to the Q6 is option B and not A.

    • Ashit - Thanks for writing. Actually, there was a typo in PS question number 5 as I specified above in my response to Marty's post.

      I just rechecked the PS question no 6 and the answer choices and there is nothing wrong with the choices. Please do try the question again and we will discuss it as needed. Thank you and all the best to you !!

  • Can you please send me the solutions?

    Promit

    • Could you send me your email address so that I can send you the solutions ?

  • Answer choice E of PS question 5 is missing something.

    • Yes, you are right Marty. The correct option is (E) x 0. Thanks for pointing out the error.

  • Also 4x^5 is written as 4^x^5 making the problem not match the answers.

    • Yes, you are right here as well. It's 4x^5 and NOT 4^x^5. Thanks again !!

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