We need to determine the correct equation that represents the relationship between two positive integers that are 3 units apart and whose product is 108. Let's break down the problem step-by-step:
1. Define the Variables:
- Let the greater integer be [tex]\( m \)[/tex].
- Since the integers are 3 units apart, the smaller integer will be [tex]\( m - 3 \)[/tex].
2. Set Up the Product Equation:
- According to the problem, the product of these two integers is 108. Therefore, the equation relating the integers and their product is:
[tex]\[
m \times (m - 3) = 108
\][/tex]
3. Examine the Given Choices:
- [tex]\( m(m+3) = 108 \)[/tex]: This equation suggests that the integers differ by 3, but in the opposite direction, making the second integer [tex]\( m + 3 \)[/tex]. This does not match the given problem where the smaller integer is [tex]\( m - 3 \)[/tex].
- [tex]\( (m+3)(m-3) = 108 \)[/tex]: This equation represents the product of the sum and difference of the same integer, which simplifies to [tex]\( m^2 - 9 \)[/tex]. This does not represent the given conditions correctly.
- [tex]\( (m-12)(m-9) = 108 \)[/tex]: This implies that the integers are [tex]\( m-12 \)[/tex] and [tex]\( m-9 \)[/tex], which are not 3 units apart.
- [tex]\( m(m-3) = 108 \)[/tex]: This directly matches our problem description where the greater integer is [tex]\( m \)[/tex] and the smaller integer is [tex]\( m - 3 \)[/tex], and their product is 108.
4. Conclusion:
- The correct equation that matches the conditions of the problem is:
[tex]\[
m(m-3) = 108
\][/tex]
Thus, the equation that can be used to solve for [tex]\( m \)[/tex], the greater integer, is:
[tex]\[
m(m-3) = 108
\][/tex]
