Let's solve this step-by-step:
### Part (a)
Given the inequality:
[tex]\[ a - b < 0 \][/tex]
To find the relationship between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
1. Start by isolating [tex]\( a \)[/tex]:
[tex]\[ a - b < 0 \][/tex]
2. Add [tex]\( b \)[/tex] to both sides:
[tex]\[ a < b \][/tex]
So, the simplified inequality that represents the direct relationship between [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is:
[tex]\[ a < b \][/tex]
### Part (b)
Given:
[tex]\[ c < 0 \][/tex]
We need to determine whether [tex]\( c(a - b) \)[/tex] is always greater than 0.
1. Consider when [tex]\( a - b > 0 \)[/tex] (i.e., [tex]\( a > b \)[/tex]):
- Since [tex]\( c \)[/tex] is less than 0 (negative), multiplying a positive number [tex]\( (a - b) \)[/tex] by a negative number [tex]\( c \)[/tex] will result in a negative number.
[tex]\[ c(a - b) < 0 \][/tex]
2. Consider when [tex]\( a - b < 0 \)[/tex] (i.e., [tex]\( a < b \)[/tex]):
- Since [tex]\( c \)[/tex] is less than 0 (negative), multiplying a negative number [tex]\( (a - b) \)[/tex] by a negative number [tex]\( c \)[/tex] will result in a positive number.
[tex]\[ c(a - b) > 0 \][/tex]
Therefore, [tex]\( c(a - b) \)[/tex] is not always greater than 0. It depends on the sign of [tex]\( (a - b) \)[/tex]. If [tex]\( a > b \)[/tex], then [tex]\( c(a - b) \)[/tex] is negative. If [tex]\( a < b \)[/tex], then [tex]\( c(a - b) \)[/tex] is positive.
In conclusion:
- The direct relationship between [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is [tex]\( a < b \)[/tex].
- [tex]\( c(a - b) \)[/tex] is not always greater than 0 because the product of a negative [tex]\( c \)[/tex] and positive [tex]\( (a - b) \)[/tex] will be negative, and the product of a negative [tex]\( c \)[/tex] and negative [tex]\( (a - b) \)[/tex] will be positive.
