Alright, let's tackle the problem step-by-step.
### Step 1: Identify the largest 5-digit number starting with 2 and ending with 8
The largest 5-digit number that begins with a 2 and ends with an 8 has the format:
[tex]\[ 2\_\_\_8 \][/tex]
To maximize this number, we need to fill the three middle digits with the highest possible values, which is 9. Thus, the number becomes:
[tex]\[ 29998 \][/tex]
### Step 2: Identify the smallest 5-digit number starting with 2 and ending with 8
Now, let's find the smallest 5-digit number that starts with a 2 and ends with an 8, which also has the format:
[tex]\[ 2\_\_\_8 \][/tex]
To minimize this number, we need to fill the three middle digits with the smallest possible values, which is 0. Thus, the number becomes:
[tex]\[ 20008 \][/tex]
### Step 3: Calculate the difference
To find the difference between these two numbers, we subtract the smallest number from the largest number:
[tex]\[
\begin{align*}
29998 - 20008 &= 9990
\end{align*}
\][/tex]
### Conclusion
The largest 5-digit number starting with 2 and ending with 8 is [tex]\( 29998 \)[/tex]. The smallest 5-digit number starting with 2 and ending with 8 is [tex]\( 20008 \)[/tex]. The difference between these two numbers is [tex]\( 9990 \)[/tex].
Thus, the answer is [tex]\( 9990 \)[/tex].
