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Which of the following is satisfied with |x-4|+|x-3|<2?

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Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post Updated on: 30 Jan 2016, 07:21
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Which of the following is satisfied with |x-4|+|x-3|<2?

A. 1<x<5
B. 2<x<5
C. 2.5<x<4.5
D. 2.5<x<4
E. 3<x<4

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Originally posted by MathRevolution on 27 Jan 2016, 18:24.
Last edited by chetan2u on 30 Jan 2016, 07:21, edited 3 times in total.
Reformatted the question.
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post 30 Jan 2016, 06:15
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Which of the following is satisfied with |x-4|+|x-3|<2?

A. 1<x<5 B. 2<x<5 C. 2.5<x<4.5 D. 2.5<x<4 E. 3<x<4


--> If there is addition when there are 2 absolute values, you can just ignore the middle. That is, |x-4|+|x-3|<2 -> |x-4+x-3|<2 -> |2x-7|<2, -2<2x-7<2, 5<2x<9, 5/2<x<9/2 -> 2.5<x<4.5
Therefore, the answer is C.
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post 27 Jan 2016, 19:01
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MathRevolution wrote:
Which of the following is satisfied with |x-4|+|x-3|<2?

A. 1<x<5 B. 2<x<5 C. 2.5<x<4.5 D. 2.5<x<4 E. 3<x<4


* A solution will be posted in two days.


Remember that |x-a| represents distance of x from 'a'. As mentioned in the question,

|x-4|+|x-3|<2 simply means that you need to find the range of 'x' such that distance of x from 4 + distance of x from 3 < 2.

Refer to the attached image, you realize that the distance 'y' when it is in between 3 and 4 will always be <2. For regions x<3 or x>4, the sum of the distances of 'y' from 3 and 4 will be <2 only when

y+1+y<2 ---> 2y+1<2 ---> y<0.5 or in other words, range of x is ---> 2.5 < x < 4.5

Attachment:
2016-01-27_20-50-35.jpg
2016-01-27_20-50-35.jpg [ 6.25 KiB | Viewed 5941 times ]


C is thus the correct answer.

Alternate solution: using POE to arrive at the correct answer.

Once you know that the sum of the distances of x from 3 and 4 must be < 2,

A. 1<x<5 . Try with x=1.5. Clearly the distance of 1.5 from 3 and 4 will be >2. Eliminate.

B. 2<x<5 . Take x=2.1. Again, the distances of 2.1 from 3 and 4 (=0.9 and 1.9 respectively) will sum to > 2. Hence Eliminate.

C. 2.5<x<4.5. Works for all possible values. Keep.

D. 2.5<x<4. Take x=4.2. The distances of 4.2 from 3 and 4 (=1.2 and 0.2 respectively) will sum to < 2 but this value is NOT in the given range, thus incomplete range. Hence Eliminate.

E. 3<x<4. Take x=2.6. It works for the given sums of the distances < 2 but is not in the given range, thus incomplete range. Hence eliminate.

C is thus the correct answer.

Hope this helps.
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post 27 Jan 2016, 18:55
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If we put x=2.51 , 2.501 ...
|x-4|+|x-3| = 1.98 , 1.998 ..... just below 2

Again if we put x= 4.49,4.499.....
|x-4|+|x-3|=1.98,1.998 ... just below 2

So Range should be
2.5<x<4.5

Answer (C)
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post 13 Feb 2016, 06:00
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Given the modules |x-4| and |x-3|, we have 2 key points (4 and 3) which give us 3 cases:

1. x<3
-x+4-x+3<2 --> \(x<\frac{5}{2}\) -->2.5<x<3 Solution is valid.

2. 3\(\leq{x}\)<4

-x+4+x-3<2 --> 1<2 Solution is valid.

3. 4\(\leq{x}\)

x-4+x-3<2 --> x\(\leq\frac{9}{2}\) --> x\(\leq4.5\) Solution is valid.

Therefore x must belong to the interval 2.5<x\(\leq{4.5}\) to satisfy the given in-equation.
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post 15 May 2017, 05:31
A. 1<x<5
B. 2<x<5
C. 2.5<x<4.5
D. 2.5<x<4
E. 3<x<4
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post 20 Sep 2018, 03:28
FelixM wrote:
Given the modules |x-4| and |x-3|, we have 2 key points (4 and 3) which give us 3 cases:

1. x<3
-x+4-x+3<2 --> \(x<\frac{5}{2}\) -->2.5<x<3 Solution is valid.

2. 3\(\leq{x}\)<4

-x+4+x-3<2 --> 1<2 Solution is valid.

3. 4\(\leq{x}\)

x-4+x-3<2 --> x\(\leq\frac{9}{2}\) --> x\(\leq4.5\) Solution is valid.

Therefore x must belong to the interval 2.5<x\(\leq{4.5}\) to satisfy the given in-equation.


Hi....

I need to learn this key point method......can you send some links please?

regards
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?  [#permalink]

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New post 20 Oct 2018, 05:55
MathRevolution wrote:
Which of the following is satisfied with |x-4|+|x-3|<2?

A. 1<x<5 B. 2<x<5 C. 2.5<x<4.5 D. 2.5<x<4 E. 3<x<4


--> If there is addition when there are 2 absolute values, you can just ignore the middle. That is, |x-4|+|x-3|<2 -> |x-4+x-3|<2 -> |2x-7|<2, -2<2x-7<2, 5<2x<9, 5/2<x<9/2 -> 2.5<x<4.5
Therefore, the answer is C.


Hello MathRevolution ,

If there is addition when there are 2 absolute values, you can just ignore the middle
Is this because lal+lbl>=la+bl ?
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Re: Which of the following is satisfied with |x-4|+|x-3|<2?   [#permalink] 20 Oct 2018, 05:55
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