Which of the following are correct?
A. If \( |x-5|\lt 2 \), then \( x\in(3,7) \)
B. If \( |x-1|\lt 4 \), then \( x\in(-3,5) \)
C. If \( |x-2|\lt 6 \) , then \( x\in(4,8) \)
D. If \( |x-4|\lt 1 \), then \( x\in(-3,5) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A and B only
For b > 0, \( |x-a|\lt b \) means \( a-b\lt x\lt a+b \).
\( \displaystyle |x-5|\lt 2\;\Rightarrow\;3\lt x\lt 7 \)
\( \displaystyle |x-1|\lt 4\;\Rightarrow\;-3\lt x\lt 5 \)
\( \displaystyle |x-2|\lt 6\;\Rightarrow\;-4\lt x\lt 8 \)
\( \displaystyle |x-4|\lt 1\;\Rightarrow\;3\lt x\lt 5 \)
So the four intervals are (3, 7), (−3, 5), (−4, 8) and (3, 5).
Checking the statements
(A) If \( |x-5|\lt 2 \), then \( x\in(3,7) \) — Correct. The interval is 5 − 2 to 5 + 2, i.e. (3, 7).
(B) If \( |x-1|\lt 4 \), then \( x\in(-3,5) \) — Correct. The interval is 1 − 4 to 1 + 4, i.e. (−3, 5).
(C) If \( |x-2|\lt 6 \) , then \( x\in(4,8) \) — Incorrect. The interval is 2 − 6 to 2 + 6, i.e. (−4, 8).
(D) If \( |x-4|\lt 1 \), then \( x\in(-3,5) \) — Incorrect. The interval is 4 − 1 to 4 + 1, i.e. (3, 5).