Which of the following is equivalent to 0

Question

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

07tiger · Answer

|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

iamaxime · Answer

| x-1 | < 1

Solve: x-1=0 -> x=1

if x> 1 then |x-1| is positive: x-1<1 and x<2
if x<1 then |x-1| is negative: -(x-1) < 1 which gives -x +1<1 and -x < 0 (multiplie times *-1 to reverse inequality) x>0

To recap we have x<2 and x>0 which gives 0<x<2 which is the inequality in the question.

So answer E.

BrushMyQuant · Answer

We need to find which of the following is equivalent to \(0 < x < 2\)

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

pikachu9 · Answer

can we solve it using mid point method?

Which of the following is equivalent to 0 < x < 2?

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