To determine the error interval for a number [tex]\( x \)[/tex] that rounds to 3.1 when rounded to one decimal place, we need to identify the smallest and largest values of [tex]\( x \)[/tex] that could round to 3.1.
### Step-by-step Process:
1. Identify the central value:
The central value, when rounded to one decimal place, is given as 3.1.
2. Determine the smallest possible value [tex]\( x \)[/tex]:
When rounding to one decimal place, any value slightly less than 3.1 but still closer to 3.1 than to 3.0 would round to 3.1. Therefore, we need to find the lowest value that rounds to 3.1.
To do this, subtract 0.05 from 3.1:
[tex]\[
\text{Lower Bound} = 3.1 - 0.05 = 3.0500000000000003
\][/tex]
3. Determine the largest possible value [tex]\( x \)[/tex]:
Similarly, for the largest value that still rounds to 3.1, we consider any value slightly below 3.15 as it would round to 3.1 when rounded to one decimal place.
Add 0.049 to 3.1:
[tex]\[
\text{Upper Bound} = 3.1 + 0.049 = 3.149
\][/tex]
4. Construct the error interval:
The error interval for the number [tex]\( x \)[/tex] is the range within which [tex]\( x \)[/tex] lies, and it will round to 3.1 when rounded to one decimal place. Therefore, [tex]\( x \)[/tex] can range from 3.0500000000000003 to 3.149.
### Final Answer:
[tex]\[
\boxed{3.0500000000000003 \leq x < 3.149}
\][/tex]
Any number [tex]\( x \)[/tex] within this interval, will round to 3.1 when rounded to one decimal place. Therefore, the error interval for [tex]\( x \)[/tex] is [tex]\( 3.0500000000000003 \leq x < 3.149 \)[/tex].
