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Around the World in 80 Questions (Day 4): How many values of x satisfy

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Bunuel
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Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:00
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How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

(A) 0
(B) 1
(C) 2
(D) 3
(E) Infinite number


 


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A
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   Updated on: 21 Jul 2023, 21:43
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Asked: How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

Case 1: x < 11
|x - 11| + |x - 12| = 11-x + 12-x< 1
23-2x < 1
2x > 22
x > 11
Not feasible since x<11

Case 2: 11<=x<12
|x - 11| + |x - 12| = x-11 + 12-x< 1
1 < 1
Not feasible

Case 3: x>=12
|x - 11| + |x - 12| = x-11 + x-12< 1
2x -23<1
2x<24
x<12
Not feasible since x>=12

IMO A

Second approach: -
|x-11|+|x-12| is the sum of distance of x from 11 and distance of x from 12
The sum of distance of x from 11 and distance of x from 12 is minimum when x lies between 11 & 12 = 1 and is NOT <1 in any case


IMO A

Originally posted by Kinshook on 20 Jul 2023, 08:10.
Last edited by Kinshook on 21 Jul 2023, 21:43, edited 1 time in total.
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:17
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Given: How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

(A) 0
(B) 1
(C) 2
(D) 3
(E) Infinite number

Solution:

Case 1: x > 0
|x - 11| + |x - 12| < 1
x - 11 + x - 12 < 1
2x - 23 < 1
2x < 24
x < 12 --------- (1)

Case 2: x < 0
|x - 11| + |x - 12| < 1
x - 11 + x - 12 > -1
2x - 23 > -1
2x > 22
x > 11 --------- (2)

From eq. (1) and (2)
11 < x < 12
Since it is not given that x is an integer, x can have decimal values as well.
Hence there can be infinite such values

Hence correct option is E
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:19
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Question : |x - 11| + |x - 12| < 1

Answer : for whatever value of x, since we are taking modulus the resultant value is always going to be positive and greater than 1.
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:20
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Even if x=11,
The value would be 1.
If x =12, value would be 1.

It is not possible to make the equation 0, and since there are mods, it will never be negative.

Therefore correct answer would be A.

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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:23
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Here no value would be lesser than 1 at the best at the lowest it would be equal to one, that's when we take x as either 11 or 12 hence answer should be A

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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:27
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How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

If x = 11 --> |0| + |-1| = 1 (not less than 1)
If x = 12 ---> |1| + |0| (not less than 1)
If x > 12 (e.g. x=13) ---> |2| + |1| = 3 (not less than 1) -- it would be the same for any value of x < 11

If x = 11.5 ---> |0.5| + |0.5| = 1 (not less than 1) -- any value of x between 11 and 12 will give us a value of 1

Therefore, 0 values satisfy the given equation and the answer is [A]
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:29
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given inquality

|x - 11| + |x - 12| < 1
case 1:
x-11+x-12 <1
2x-23<1
2x<24
x<12

case 2:
-x+11-x+12<1
-2x+23<1
-2x<-22
2x>22
x>11
range of x
11<x<12

option A ; 0


Bunuel wrote:
How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

(A) 0
(B) 1
(C) 2
(D) 3
(E) Infinite number


 


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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:30
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|x-11|+|x-12|<1
f(x)=abs(x-11)+abs(x-12) has min value of 1. hene no solution
A)
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:33
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we can read the question as distance between x and 11 + distance between x and 12 < 1

if you plot x on numberline you can visualize it's not possible distance will always be greater than 1

20 sec ans 0

ans A
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:36
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My Answer: Option A, 0

There can be 4 different scenarios for this inequality. For (x-11) and (x-12) either both can be taken as positive, both can be taken as negative or one can be taken as positive and the other as negative.
Now, if one is taken as positive and the other is taken as negative, x will cancel out and hence, these 2 scenarios are invalid.

Now, if both are taken as positive,
x-11+x-12<1 , x<12

if both are taken as negative,
-x+11-x+12<1, x>11

Since there is no overlap between x<12 and x>11, 0 values of x satisfy the inequality.
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:38
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|x - 11| + |x - 12| < 1

Case 1 : x<11
11-x + 12-x <1
x>11
No solution

Case 2 :11<x<12
x-11 +12-x<1
1<1
No solution

Case 3: x>12
x-11 + x-12<1
x<12

No solution

Therefore 0 solutions
Answer A.
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:39
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Bunuel wrote:
How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

(A) 0
(B) 1
(C) 2
(D) 3
(E) Infinite number


 


This question was provided by GMAT Club
for the Around the World in 80 Questions

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|x - 11| + |x - 12| < 1 implies that the sum of the distances of x from 11 and 12 on the number line is less than 1.
However, the minimum possible sum of these distances is 1, so there are no values of x that satisfy the inequality
Answer = 0
A
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:47
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How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?


try putting number with small margin suppose x = 11.1
l11.1-11l =0.1
l11.1-12l = 0.9
hence 0.1+0.9 =1
so no value
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:51
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Bunuel wrote:
How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

(A) 0
(B) 1
(C) 2
(D) 3
(E) Infinite number


 


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for the Around the World in 80 Questions

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------ REGION 1 ------- 11 --- REGION 2 --------- 12 ---------- REGION 3

REGION 1

-(x-11)-(x-12) < 1
-2x < -22
x > 11

Ignore this region

REGION 2

(x-11)-(x-12) < 1
12 - 11 < 1
1 < 1

Ignore this region

REGION 3

x - 11 + x - 12 < 1
2x - 23 < 1
x < 12

Ignore this region

No value satisfy.

IMO A
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 08:53
The answer is C i.e 2
Ix-11I + Ix-12I < 1 leads to x<12
and x>-11 when taking the negative sign
Hence, -11<X<12
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 09:06
How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

(A) 0
(B) 1
(C) 2
(D) 3
(E) Infinite number


x=11, 12

case I: x>12
x-11 + x-12<1
2x<24
x<12

case II: 11<x<12
x-11 + x-12<1
x<12

case III: x<11
-x+11 -x+12 <1
-2x + 23<1
-2x<-22
x>11

thus the 11<x<12 - thus infinite values
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 09:08
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How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?

x=11 then we get 1 , but 1 is not less than 1. Similarly for 12. So zero values satisfy it.

A
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 09:24
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How many values of x satisfy the inequality |x - 11| + |x - 12| < 1?
___________________________________
In solving the question , we can basically put numbers instead of x.
So, it can be integer or decimal number such as 11.6

If we replace x with 12, 11 , or 13 , the inequality does not hold. Because , we get absolute value from modules.
If we replace x with 11.6 , 11.98 or any number between 11 and 12, the inequality still doesnt hold.

So, asnwer is A.
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink] New post   20 Jul 2023, 09:29
|x - 11| + |x - 12| < 1

Both +
x - 11 + x - 12 < 1
2x - 23 < 1
x < 12

Both -
-x + 11 - x + 12 < 1
-2x + 23 < 1
-2x < -22
x > 11

There are infinite numbers between 11 and 12
Ans: E
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Re: Around the World in 80 Questions (Day 4): How many values of x satisfy [#permalink]
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