D01-14

Official Solution:

If \(-1 < x < 5\) then which of the following must be true?

A. \(|3 - x| \lt -3\)
B. \(|x| \lt 4\)
C. \(|x| - 2 \gt 2\)
D. \(|2 + x| \gt 3\)
E. \(|x - 2| \lt 3\)


Given that \(-1 < x < 5\), we need to determine which of the options is always true.

A. \(|3-x| < -3\). This option is always false since the absolute value of a number is always non-negative.

B. \(|x| < 4\). This option is not always true since it implies that \(-4 < x < 4\), which does not hold for all values of \(x\). For example, if we take \(x=4.5\), this inequality is not satisfied.

C. \(|x| - 2 > 2\). This option is also not always true. It simplifies to \(|x| > 4\), which means that \(x < -4\) or \(x > 4\). But for \(x=0\), this inequality is not satisfied.

D. \(|2 + x| > 3\). This option is also not always true. It gives \(2 + x > 3\) or \(2 + x < -3\), which simplifies to \(x > 1\) or \(x < -5\), but for \(x=0\), this inequality is not satisfied.

E. \(|x-2| < 3\). This means that \(-3 < x - 2 < 3\). Adding 2 to each part gives \(-1 < x < 5\), which is already given in the stem. Hence, this option is always true.

Therefore, the only option that must be true for all values of \(x\) in the given range is option E.


Answer: E