To solve this problem, let's clearly define what it means for a number to be odd and how we can determine the largest odd number that precedes a given odd number [tex]\( x \)[/tex].
1. Understanding Odd Numbers:
- An odd number is any integer that cannot be exactly divided by 2.
- Some examples of odd numbers are 1, 3, 5, 7, 9, etc.
2. Identifying Preceding Odd Number:
- To identify the largest odd number preceding [tex]\( x \)[/tex], we need to find the odd number that is immediately before [tex]\( x \)[/tex].
3. Analyzing Choices:
- Let's go through each given option to see which one fits our requirement.
Option (a) [tex]\( x - 1 \)[/tex]:
- If [tex]\( x \)[/tex] is an odd number, [tex]\( x - 1 \)[/tex] would be an even number.
- For example, if [tex]\( x = 5 \)[/tex], then [tex]\( 5 - 1 = 4\)[/tex], which is even. This option is incorrect.
Option (b) [tex]\( x - 2 \)[/tex]:
- If [tex]\( x \)[/tex] is an odd number, subtracting 2 from it will still yield an odd number.
- For example, if [tex]\( x = 5 \)[/tex], then [tex]\( 5 - 2 = 3\)[/tex], which is both smaller than 5 and odd.
- Similarly, if [tex]\( x = 7 \)[/tex], then [tex]\( 7 - 2 = 5 \)[/tex], which is odd as well.
- This option fits our requirement.
Option (c) [tex]\( x - 3 \)[/tex]:
- If [tex]\( x \)[/tex] is an odd number, then [tex]\( x - 3 \)[/tex] will also be an even number.
- For example, for [tex]\( x = 5 \)[/tex], [tex]\( 5 - 3 = 2\)[/tex], which is even. This option is incorrect.
Option (d) [tex]\( x - 4 \)[/tex]:
- If [tex]\( x \)[/tex] is an odd number, subtracting 4 will keep the result odd.
- However, [tex]\( x - 4 \)[/tex] is not the largest odd number preceding [tex]\( x \)[/tex]; it skips over one odd number to yield another.
- For instance, if [tex]\( x = 5 \)[/tex], then [tex]\( 5 - 4 = 1 \)[/tex]. While 1 is odd, 3 (which is [tex]\( x - 2 \)[/tex]) is larger than 1.
- This option does not find the immediately preceding odd number but rather another odd number.
Conclusion:
The correct option is:
[tex]\[ \text{(b) } x - 2 \][/tex]
So, the answer is:
[tex]\[ \boxed{x - 2} \][/tex]
