Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Example #2 Q.: \(|x^2-4| = 1\). What is x? Solution: There are 2 conditions:

a) \((x^2-4)\geq0\) --> \(x \leq -2\) or \(x\geq2\). \(x^2-4=1\) --> \(x^2 = 5\). x e {\(-\sqrt{5}\), \(\sqrt{5}\)} and both solutions satisfy the condition.

b) \((x^2-4)<0\) --> \(-2 < x < 2\). \(-(x^2-4) = 1\) --> \(x^2 = 3\). x e {\(-\sqrt{3}\), \(\sqrt{3}\)} and both solutions satisfy the condition.

Hello, Nice post. I can't understand why the first condition equals to 0 as well when the second condition is only less than zero but NOT equal to zero? Could you please explain?

Thanks! :D

For modulus or absolute value questions, you need to take the equality with the '>' sign. That is the convention and follows the definition of an absolute value. Additionally, you only need to account for equality once in your question.

Also, think of modulus or absolute value this way:

|x| = x for x=0,1,2,3,4,5... or true for all NON-NEGATIVE numbers. Remember the 'non-negative' part. This includes 0 as well.

But |x| = -x for x<0, x=-1,-2,-0.25 ...

This is the reason why we put equality with the '>' sign to account for all non-negative numbers. The 'nature' of 'x' does not change if it is 0 and above but it does change (multiply x by -1) if x is <0.

Hope this helps.

Thanks. Now I'm observing the => convention |x| has. The equal sign is included because we are guessing x could be 0 as well. But if 0 is a non-negative number, it can be a non-positive number too (In that non positive number scenario, we would have followed the <= convention for |x|). In fact 0 is a neutral number. So, in this case, why is 0 implied as a non negative number?
_________________

For modulus or absolute value questions, you need to take the equality with the '>' sign. That is the convention and follows the definition of an absolute value. Additionally, you only need to account for equality once in your question.

Also, think of modulus or absolute value this way:

|x| = x for x=0,1,2,3,4,5... or true for all NON-NEGATIVE numbers. Remember the 'non-negative' part. This includes 0 as well.

But |x| = -x for x<0, x=-1,-2,-0.25 ...

This is the reason why we put equality with the '>' sign to account for all non-negative numbers. The 'nature' of 'x' does not change if it is 0 and above but it does change (multiply x by -1) if x is <0.

Hope this helps.

Thanks. Now I'm observing the => convention |x| has. The equal sign is included because we are guessing x could be 0 as well. But if 0 is a non-negative number, it can be a non-positive number too (In that non positive number scenario, we would have followed the <= convention for |x|). In fact 0 is a neutral number. So, in this case, why is 0 implied as a non negative number?

Please refer to the text in red above. The nature of |x| does not change when x = 0 or >0 while it changes when x <0. This is the reason why we club =0 with >0 as they have similar "nature".

Thank you for the quick and clear reply - this makes so much sense now! Basically |x| = -x when |x| < 0; |x| = x when |x| > or = 0, so I need to test each of the |x+3|, |4-x| and |8+x| to see if they are > 0 or <0 and translate them accordingly to + or - (x+3), (4-x) etc. Is this the correct understanding?
_________________

Working towards 25 Kudos for the Gmatclub Exams - help meee I'm poooor

Thank you for the quick and clear reply - this makes so much sense now! Basically |x| = -x when |x| < 0; |x| = x when |x| > or = 0, so I need to test each of the |x+3|, |4-x| and |8+x| to see if they are > 0 or <0 and translate them accordingly to + or - (x+3), (4-x) etc. Is this the correct understanding?

a) x<−8. −(x+3)−(4−x)=−(8+x) --> x=−1. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) −8≤x<−3. −(x+3)−(4−x)=(8+x) --> x=−15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) −3≤x<4. (x+3)−(4−x)=(8+x) --> x=9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x≥4. (x+3)+(4−x)=(8+x) --> x=−1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

I have a question about the above. Why is there a negative (-) sign in front of the x+3 term in part a), and not in front of the 4-x term? I think this gets at some basics of absolute values, but am not sure. Any insight is much appreciated. Thanks.

Edit: see question asked right before me, never mind.

Hi, Walker. many thanks for your great explanation but I think I need a further explanation for stupid person like me. how did you decide the sign of the three expressions in the four cases. please note that I may miss some advanced concepts of absolute value. I need like a map of what are the kind of absolute value question and what are the best and fastest approaches to solve each kind. please help. I need to master my self in absolute value questions especially I have like 2 month left to take the test.

Hi, Walker. many thanks for your great explanation but I think I need a further explanation for stupid person like me. how did you decide the sign of the three expressions in the four cases. please note that I may miss some advanced concepts of absolute value. I need like a map of what are the kind of absolute value question and what are the best and fastest approaches to solve each kind. please help. I need to master my self in absolute value questions especially I have like 2 month left to take the test.

In order for anyone to answer your question to the best of their abilities, make sure to quote the question and the step that you have a doubt about. A general question such as the one above will not serve you any good. Please re post your question quoting a particular question from the 1st page of this topic and then we can look into your specific question.

Can u please help? Is that correct that I could solve the 3-mod equation/inequality by opening mod one by one and/or in different combinations (negative-positive) and then test the resulting values OR by setting the intervals and testing values out of those intervals to see whether they suffice the requirements set by the intervals AND get the same result?

The reason for intervals is that they save time, while dealing with 3-mod questions can become a time-consuming exercise) and prone to errors.

I have been reading this from GMAT Club Math Book and have been trying to understand the 3-step process to solving absolute value inequalities/equalities and vice versa. I cannot seem to understand how to use this approach.

If anyone can explain it in more elaborate detail I would be very greatful.

I need a small help in explaining a few bits to me in a little detail (I am somehow missing something here to get it all right). Can anyone please help. TIA

1) In the tricks example I please explain this bit in a little detail- "Now, let’s look at our options. Only B and D has 8/2=4 on the right side and D had left site 0 at x=5. Therefore, answer is D."

2) In the trick no II, please explain the second example- "|x+3|>3 is equal to x e (-inf,-6)&(0,+inf)"

sorry if its a very basic query, just want to get the basics right

I'm having problems understanding the example 1 ( as written below). Why are they key points -8, -3 and 4, instead of 3, 4, 8?

Many thanks

Example #1 Q.: |x+3|−|4−x|=|8+x||x+3|−|4−x|=|8+x|. How many solutions does the equation have? Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) x<−8x<−8. −(x+3)−(4−x)=−(8+x)−(x+3)−(4−x)=−(8+x) --> x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) −8≤x<−3−8≤x<−3. −(x+3)−(4−x)=(8+x)−(x+3)−(4−x)=(8+x) --> x=−15x=−15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) −3≤x<4−3≤x<4. (x+3)−(4−x)=(8+x)(x+3)−(4−x)=(8+x) --> x=9x=9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x≥4x≥4. (x+3)+(4−x)=(8+x)(x+3)+(4−x)=(8+x) --> x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

I'm having problems understanding the example 1 ( as written below). Why are they key points -8, -3 and 4, instead of 3, 4, 8?

Many thanks

Example #1 Q.: |x+3|−|4−x|=|8+x||x+3|−|4−x|=|8+x|. How many solutions does the equation have? Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) x<−8x<−8. −(x+3)−(4−x)=−(8+x)−(x+3)−(4−x)=−(8+x) --> x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) −8≤x<−3−8≤x<−3. −(x+3)−(4−x)=(8+x)−(x+3)−(4−x)=(8+x) --> x=−15x=−15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) −3≤x<4−3≤x<4. (x+3)−(4−x)=(8+x)(x+3)−(4−x)=(8+x) --> x=9x=9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x≥4x≥4. (x+3)+(4−x)=(8+x)(x+3)+(4−x)=(8+x) --> x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

|x - a| = b means x is b units away from a.

|x + a| = b |x - (-a)| = b means x is b units away from -a.

Hence, if we have |x + 3|, it means the transition point is -3.

We’ve given one of our favorite features a boost! You can now manage your profile photo, or avatar , right on WordPress.com. This avatar, powered by a service...

Sometimes it’s the extra touches that make all the difference; on your website, that’s the photos and video that give your content life. You asked for streamlined access...

“Keep your head down, and work hard. Don’t attract any attention. You should be grateful to be here.” Why do we keep quiet? Being an immigrant is a constant...

“Keep your head down, and work hard. Don’t attract any attention. You should be grateful to be here.” Why do we keep quiet? Being an immigrant is a constant...