D01-14

Question

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

Bunuel · Accepted Answer

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

joe123x · Answer

I have a simple question, I got this right but by substituting numbers.

Q: in your explanation how does |x-2|<3 turn to -3<x-2<3 ?

|x-2| can be treated as x-2 or -x+2, right?

Bunuel · Answer

Consider this, when we have the inequality |a| < 2, it means that the distance between  a  and 0 is less than 2:

-----(-2) ------0------ 2-----

Therefore, |a| < 2 implies that -2 < a < 2.

Similarly, when we have the inequality |x - 2| < 3, it means that -3 < x - 2 < 3.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

kilicon · Answer