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Which of the following is equivalent to 0 < x < 2 ?
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07 Nov 2019, 11:34
07 Nov 2019, 11:34
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Question Stats:
66% (00:53) correct
34% (00:44) wrong based on 50 sessions
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Which of the following is equivalent to \(0 < x < 2\)?
A. \(x=1\)
B. \(|x|<1\)
C. \(|x|<2\)
D. \(|x+1|<1\)
E. \(|x-1|<1\)
A. \(x=1\)
B. \(|x|<1\)
C. \(|x|<2\)
D. \(|x+1|<1\)
E. \(|x-1|<1\)
Re: Which of the following is equivalent to 0 < x < 2 ?
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20 Apr 2020, 15:58
20 Apr 2020, 15:58
Expert Reply
|x-1|<1
1. (x-1)<1
= x<1+1 = x<2
2. -(x-1)<1
= -x+1<1
= -x<1-1 = -x<0 = x>0
Combining workings 1 and 2 yields: 0<x<2 which is equal to the expression in the question '0<x<2'
Answer: E
1. (x-1)<1
= x<1+1 = x<2
2. -(x-1)<1
= -x+1<1
= -x<1-1 = -x<0 = x>0
Combining workings 1 and 2 yields: 0<x<2 which is equal to the expression in the question '0<x<2'
Answer: E
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Which of the following is equivalent to 0 < x < 2 ?
[#permalink]
16 Jul 2021, 02:34
16 Jul 2021, 02:34
Let's look at two ways of solving the problem
Method 1: Algebra
Let's solve each answer choice
A. \(x=1\)
Now 0 < x < 2. Although x=1 lies in the range but it does not cover the entire range as x can be decimal also like 1.1 => FALSE
B. \(|x|<1\)
|x| < 1 => -1 < x < 1 (because |x| < a => -a < x < a) => FALSE
C. \(|x|<2\)
|x| < 2 => -2 < x < 2 (because |x| < a => -a < x < a) => FALSE
D. \(|x+1|<1\)
|x+1| < 1 => -1 < x+1 < 1 (because |x| < a => -a < x < a)
Subtract 1 from all the sides we get
-1-1 < x+1 -1 < 1-1
=> -2 < x < 0 which is not equivalent to 0 < x < 2 => FALSE
E. \(|x-1|<1\)
|x-1| < 1 => -1 < x-1 < 1 (because |x| < a => -a < x < a)
Add 1 to all the sides we get
-1+1 < x-1 +1 < 1+1
=> 0 < x < 2 => TRUE
So, Answer will be E
Method 2: Substitution (Eliminate answer choices)
Let's take two values of x to check. x = 1 and x=1.5
A. \(x=1\) -> 1.5 is not satisfied. This is anyways a value and we are looking for a range
B. \(|x|<1\) -> x=1 is not satisfied as |1| will be equal to 1 and not < 1
C. \(|x|<2\) -> both x=1 and x=1.5 satisfy but x = -1 also satisfies this. Which is not in our range of \(0 < x < 2\)
D. \(|x+1|<1\) -> x=1 makes this \(|1+1|<1\) which is not true
E. \(|x-1|<1\) -> both x=1 and x=1.5 satisfy this and this is the only choice left now.
So, Answer will be E
Hope it helps!
To learn more about Absolute value, watch the following video
Method 1: Algebra
Let's solve each answer choice
A. \(x=1\)
Now 0 < x < 2. Although x=1 lies in the range but it does not cover the entire range as x can be decimal also like 1.1 => FALSE
B. \(|x|<1\)
|x| < 1 => -1 < x < 1 (because |x| < a => -a < x < a) => FALSE
C. \(|x|<2\)
|x| < 2 => -2 < x < 2 (because |x| < a => -a < x < a) => FALSE
D. \(|x+1|<1\)
|x+1| < 1 => -1 < x+1 < 1 (because |x| < a => -a < x < a)
Subtract 1 from all the sides we get
-1-1 < x+1 -1 < 1-1
=> -2 < x < 0 which is not equivalent to 0 < x < 2 => FALSE
E. \(|x-1|<1\)
|x-1| < 1 => -1 < x-1 < 1 (because |x| < a => -a < x < a)
Add 1 to all the sides we get
-1+1 < x-1 +1 < 1+1
=> 0 < x < 2 => TRUE
So, Answer will be E
Method 2: Substitution (Eliminate answer choices)
Let's take two values of x to check. x = 1 and x=1.5
A. \(x=1\) -> 1.5 is not satisfied. This is anyways a value and we are looking for a range
B. \(|x|<1\) -> x=1 is not satisfied as |1| will be equal to 1 and not < 1
C. \(|x|<2\) -> both x=1 and x=1.5 satisfy but x = -1 also satisfies this. Which is not in our range of \(0 < x < 2\)
D. \(|x+1|<1\) -> x=1 makes this \(|1+1|<1\) which is not true
E. \(|x-1|<1\) -> both x=1 and x=1.5 satisfy this and this is the only choice left now.
So, Answer will be E
Hope it helps!
To learn more about Absolute value, watch the following video
