Question

|x - 3| + |x - 2| = 1

Answer 1

x=2

Answer 2

-(x-3)+(-1(x-2))=1

(x-3)+(x-2)=1

solve for x in both equations u will gt the possible values for x :)

Answer 3

there are an infinite number of answers

Answer 4

that is correct. you have to check the 3 regions, x<2, 2<x<3, and x>3. It turns out if 2<x<3, then it is a solution.

Answer 5

x = 2 is valid, x = 2.1 is valid, x = 2.2 is valid, x = ... x = 3 is valid. any x between x = 2 and x = 3 works. question is how to arrive at this conclusion

Answer 6

If x is greater than 2, then \[|x-2|=x-2\]
if x is less than 3, then:\[|x-3|=-(x-3)=-x+3\]

Answer 7

So for 2<x<3 we get:\[|x-2|+|x-3|=x-2-(x-3)=3-2=1\]

Answer 8

well who thinks i'm incorrect ??

Answer 9

not incorrect, but incomplete.

Answer 10

there is a set of solutions. its the set [2,3]. any x such that:\[2\le x \le 3\]is a solution.

Answer 11

knowing this is it possible to simplify

\[\sqrt{x + 3 - 4\sqrt{x - 1}} + \sqrt{x + 8 - 6\sqrt{x - 1}} = 1\]

using substitution ?

Answer 12

if 'x' belong to Integer, only then there are 2 solutions, x=2,3

Answer 13

This is the condition for absolute values :-

When the value is unknown in the mod brackets we have to put the value in both positive & negative conditions.

since |-x|=-(-x) & |x|=x

Answer 14

\(x+3 -4 \sqrt{x-1} = (\sqrt{x-1})^2-2*2*\sqrt{x-1}+2^2\)

Answer 15

=\([\sqrt{x-1}+2]^2\)

Answer 16

How did you see that o.O

Answer 17

because x+3 = x-1 +4 ...

Answer 18

lol

Answer 19

similarly tr for x+8-6 sqrt{x-1}

Answer 20

hint : x+8 = x-1+9 :P

Answer 21

very clever!

Answer 22

[5, 10] ?

Answer 23

Hmm... If all answers are between 2 and 3 are there still an infinite amount of answers or would the fact that everything outside of 2 and 3 mean it has to be finite?

Answer 24

\(|\sqrt{x-1}-2|+|\sqrt{x-1}-3|=1 \\ y=\sqrt{x-1}\)

Answer 25

if one circle is circumscribed about the other, does it contain more points because it has a greater circumference? (not really a question)

Answer 26

thank you hartn, the solution was clear to me at your second post

Answer 27

ok... circle circumscribed about other means ?
circle inside a circle ?

Answer 28

maybe the wrong terminology was used. consider one circle larger than the other but with the same center. does the outer circle contain more points than the inner because it has a greater circumference? the answer is no for a simple reason

Answer 29

the outer circle will contain more points than the inner because it has a greater area.

Answer 30

@binarymimic The question is are there infinite solutions to this problem? If so how is considered infinite but yet having the boundaries between 2 and 3...

Answer 31

@patdistheman there are infinite numbers between 2 and 3.

Answer 32

There is something intuitively that seems wrong about that. All the numbers fall between 2 and 3. We know there are numbers outside of the range then wouldn't by definition the answers not be infinite?

Answer 33

Limitless or endless in space but also between 2 and 3 seem to be a contradiction.

Answer 34

all of the numbers between 2 and 3 can be put in a 1 to 1 correspondence with the natural numbers, yes?

Answer 35

infinite does not necessarily mean from -infinity to infinity , there are infinite numbers between 2 and 2.1 also.

Answer 36

1 to 1 correspondence?

Answer 37

for each number between 2 and 3 can you map it to a number in N ? where N = 1, 2, 3, ...

Answer 38

Between 2 and 2.1 would have a limited supply no?

Answer 39

perhaps not even a 1 to 1 correspondence with the natural numbers if we include irrationals, which are abundant.

Answer 40

think of it this way. for every two positive numbers, their average is between them. and this number is also positive. you can do this forever if you like and continue to obtain numbers between 2.0 and 2.1

Answer 41

It is clear that there appears to be an infinite supply of numbers between 2 and 3 that can be generated because additional numbers can always be added after but if that is infinite how would you describe the numbers outside of 2 and 3 while describing that case?

Answer 42

i would say there are an infinite number of solutions, and an infinite number of non-solutions

Answer 43

Putting a limit on infinite seems contradictory...

Answer 44

its a paradox called continuity. it is kind of a problem but we tend to overlook it.

Answer 45

Well I don't see the productive value at the moment. All I can say is it doesn't sit right.

Answer 46

If numbers are used to describe something I imagine there is a limit at which the highest number of anything can be reached. Even all the atoms in the universe have a number...

Answer 47

How would the difference between numbers that can relate to something and numbers that can not be defined...

Answer 48

For example lets say there is 10^82 atom in the observable universe... How would you describe numbers beyond something that they can be applied to...

Answer 49

well in mathematical space, how many points are there on the real line, for example? or on the plane, or in R3? etc. you reach a point at which you cannot define infinity by how many points it contains, per se, but by how the set containing it can be compared to other sets

Answer 50

I suppose is essence the numbers have no point if they can not be associated to anything... I like the description... infinite solution and infinite non solution... I see no results on google... Was that just invented? lol

Answer 51

you could argue that the region between 2 and 3 on the real line contains just as many points as R100 or R 10^83 or any finite value of n for Rn

Answer 52

? i guess. i have no other way of describing it other than using interval notation.

Answer 53

That doesn't sound correct either. I do not see why you would not be able to just keep adding zeros indefinitely. Yea I think we have something there. That may be a new form of description. How relevant that is to anything not sure.

Answer 54

it is completely paradoxical to say that a part of a line contains just as many points as the entire line does, and also just as many points as a plane does, or all of 3D space, or all of any nth dimensional space.

Answer 55

You heard it here folks... Openstudy.com... "Infinite Non Solution Numbers"

Answer 56

lol

Answer 57

I do not see how that is possible. I think the error is in thinking of mathematics as anything more than a tape measure.

Answer 58

It would stand to reason that the distance between 1 to 2 is the same the distance between 2 to 3.

Answer 59

You could have a tape measure with a length of 10^82 feet but what would you ever even do with it...