To determine the error interval for the number [tex]\( x \)[/tex] rounded to 2 significant figures as 1800, we need to find the range within which [tex]\( x \)[/tex] lies when it is rounded to two significant figures.
1. Identify the number of digits to consider for determining the rounding error:
Since 1800 is rounded to 2 significant figures, the two significant figures we are considering are 1 and 8. The last significant figure (8) is in the hundreds place, so we are rounding to the nearest hundred.
2. Determine the rounding interval:
When rounding to two significant figures, we focus on the place value represented by the second significant figure and the place value just one place lower. For 1800 (with the second significant figure being in the hundreds place), we are dealing with numbers in the range of hundreds, which means our interval boundaries depend on that last significant digit at the hundreds place.
3. Calculate the lower bound:
The lower bound is the smallest number that would round up to 1800 when considering the rounding rules. The boundary just below 1800 that rounds up to 1800 is halfway between 1700 and 1800, which is:
[tex]\[
1800 - 50 = 1750
\][/tex]
4. Calculate the upper bound:
The upper bound is the largest number that would still round down to 1800. The boundary just above 1800 that still rounds down to 1800 is halfway between 1800 and 1900, which is:
[tex]\[
1800 + 50 = 1850
\][/tex]
So, the error interval for [tex]\( x \)[/tex] is:
[tex]\[
1750 \leq x < 1850
\][/tex]
This means that [tex]\( x \)[/tex] can be any number from 1750 up to, but not including, 1850.
