Question

A salesperson’s total earnings consist of a base salary of x dollars per year, plus commission earnings of 11% of the total sales the salesperson makes during the year. This year, the salesperson has a goal for the total earnings to be at least 3 times and at most 4 times base salary. What inequalities represent all possible values of total sales s, in dollars, the salesperson can make this year in order to meet that goal?

Answer 1

To meet their earnings goal, a salesperson's total sales must fall between ≈2x/0.11 and ≈3x/0.11, inclusive. This is found by setting up inequalities that represent at least 3 times and at most 4 times their base salary and solving for the total sales.

Step-by-step explanation:

The salesperson’s total earnings can be represented by the equation E = x + 0.11s, where E is the total earnings, x is the base salary, and s is the total sales. To meet the goal that the total earnings are at least 3 times and at most 4 times the base salary, we can set up two inequalities:

1. For the lower bound (at least 3 times the base salary): 3x ≤ x + 0.11s or 2x ≤ 0.11s
2. For the upper bound (at most 4 times the base salary): 4x ≥ x + 0.11s or 3x ≥ 0.11s

To find the range of total sales s that meet the goal, solve each inequality for s:

• To solve 2x ≤ 0.11s, divide both sides by 0.11 to get s ≥ ≈2x/0.11.
• To solve 3x ≥ 0.11s, divide both sides by 0.11 to get s ≤ ≈3x/0.11.

These expressions represent all possible values of total sales s that the salesperson can make to meet their earnings goal.