Please help me with the following:
Which of the following expressions is equivalent to |x – 2| < 3?
A. –3 > x – 2 < 3
B. –3 < x – 2 < 3
C. x – 2 < 3 and x – 2 < –3
D. x – 2 < 3 or x – 2 < –3
I know the answer isn't A because I got it wrong, but I think the answer is B. Is B correct?
Please help me by checking my answer. Any help will be appreciated! :)
B is correct.
|x-2| means the difference between x and 2. That difference must be less than 3, so
2-3 < x < 2+3
-3 < x-2 < 3
Thank you Steve!! I was stuck on this one for so long! I have such a hard time with these, so it's going to happen for me. :)
To determine which expression is equivalent to |x – 2| < 3, let's break it down step by step.
The inequality |x – 2| < 3 represents the distance between x and 2 on the number line being less than 3.
First, we need to identify the two cases:
1) x – 2 is positive: In this case, |x – 2| will be equal to x – 2. So, x – 2 < 3.
2) x – 2 is negative: In this case, |x – 2| will be equal to -(x – 2), which can be rewritten as -x + 2. So, -x + 2 < 3.
Now, let's find the correct answer option:
A. –3 > x – 2 < 3:
This option doesn't consider the case when x – 2 is negative, so it's incorrect.
B. –3 < x – 2 < 3:
This option covers both cases by stating that x – 2 is greater than -3 and less than 3. Therefore, it is a possible correct answer.
C. x – 2 < 3 and x – 2 < –3:
This option mistakenly assumes that both cases occur simultaneously, which is not true. So, it's incorrect.
D. x – 2 < 3 or x – 2 < –3:
This option states that either x – 2 is less than 3 or x – 2 is less than -3. It covers both cases separately, so it's a possible correct answer.
In conclusion, both options B and D are correct expressions that are equivalent to |x – 2| < 3. Thus, your intuition was correct, and B is indeed a valid answer option. However, D is also correct, so either option B or D can be chosen as the correct answer.