Math: Absolute value (Modulus)

So...just to clarify...the answer can either be 5 or -3?

Just need the clarification on what this is saying

In the following example of 3-steps approach for complex problems


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) Can any one please explain me how x<-8 because whenever I am doing it I am getting x>=-8 for both |8+x| and -|8+x|

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)

IN Example #1
Q.: |x+3|−|4−x|=|8+x||x+3|−|4−x|=|8+x|. How many solutions does the equation have?
C condition has typo error
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.)

its not -15 but 9

chetan2u Hi, sorry I keep bugging you but your explanations are easiest to understand
Can you explain to me the "trick" mentioned in the post? I don't get it :/ What are we doing after finding the mid-point?

Please mention the portion you are talking about..
No problems. Pl keep asking questions and I would reply to each whenever I get time

"Problem: 1<x<9. What inequality represents this condition?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5
Solution: 10sec. Traditional 3-steps method is too time-consume technique. First of all we find length (9-1)=8 and center (1+8/2=5) of the segment represented by 1<x<9. 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."

This is the "trick" mentioned in the post, don't know what it means chetan2u

I didn't understand how the sign for each expression was determined.
I'd appreciate any help.
Thanks!!

When are we supposed to use the 3-step method? I mean, on which kind of problems?

OP or Bunuel

There is a typo in the first post.

It says "(-15 is not within (-3,4) interval." for the range in (c) when it should actually say "9 is not within the -3<x<4 range"

____________________
Edited. Thank you.

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I have a doubt for this question

Does not this expression mean that x^2 is at a distance of 1 from 4. Then should not the correct answers be only sq root 5 and sq root 3 instead of -sq root 3 and -sq root 5 as these don't have a distance of 1 from 4 ?

Regards

You said in each range signs of the terms will be different . I am unable to assign proper signs to terms in ranges VeritasKarishma please help

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Vini07,

It means "x^2" is at a distance 1 from 4, not x.
So x^2 is 5 or 3. Then x can be -sqrt(5) and also sqrt(5) and -sqrt(3) and also sqrt(3)

The rightmost region is positive and you alternate from there (assuming the factors are in (ax + b) or (ax - b) form). Check out the logic explained in this video:

https://youtu.be/PWsUOe77__E

____________________
Fixed. Thank you.

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No, the comparison sign gets reversed in |X-Y|

Ex,
|X+Y| <= |X|+|Y| - Less than equals

|X-Y| >= |X|-|Y| - Greater than equals