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: AlgebraLet'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 EMethod 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 EHope it helps!
To learn more about Absolute value, watch the following video _________________