Final answer:
The correct system of inequalities to model Kate's situation is option C, which expresses that the number of jars of pickles and relish combined cannot exceed 50, and the sales must be at least $260. Options D and E can be checked by substituting the values into the inequalities.
Step-by-step explanation:
In order to model the scenario with a system of inequalities for Kate's situation, we have to consider two main constraints: the number of jars she can produce and the amount of money she needs to make.
The first inequality, p + r ≤ 50, represents the limit on the maximum number of jars of pickles (p) and relish (r) Kate can make based on her cucumber supply, which cannot exceed 50 jars in total.
The second inequality, 6p + 4r ≥ 260, represents the minimum amount of money, $260, she needs to make from selling the jars. Since pickles sell for $6 per jar and relish sells for $4 per jar, this inequality makes sure Kate's sales meet or exceed her financial goal.
Therefore, looking at the given options, option C is the correct representation of the situation with the inequalities p + r ≤ 50 and 6p + 4r ≥ 260.
Regarding option D (p = 32 and r = 18) and option E (p = 28 and r = 22), we can check if these pairs are solutions by substituting them into the inequalities. If they satisfy both inequalities, they are solutions to the system.