Linear inequalities in one or two variables SAT question

Choice B is correct. It’s given that 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. If the salesperson makes $s$ dollars in total sales this year, the salesperson’s total earnings can be represented by the expression $x+0.11s$. It’s also given that the salesperson has a goal for the total earnings to be at least $3$ times and at most $4$ times the base salary, which can be represented by the expressions $3x$ and $4x$, respectively. Therefore, this situation can be represented by the inequality $3x\le x+0.11s\le 4x$. Subtracting $x$ from each part of this inequality yields $2x\le 0.11s\le 3x$. Dividing each part of this inequality by $0.11$ yields $\frac{2}{0.11}x\le s\le \frac{3}{0.11}x$. Therefore, the inequality $\frac{2}{0.11}x\le s\le \frac{3}{0.11}x$ represents all possible values of total sales $s$, in dollars, the salesperson can make this year in order to meet their goal.

Choice A is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least $2$ times and at most $3$ times, rather than at least $3$ times and at most $4$ times, the base salary.

Choice C is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least $3$ times and at most $4$ times the base salary.

Choice D is incorrect. This inequality represents a situation in which the total earnings are at least $4$ times and at most $5$ times, rather than at least $3$ times and at most $4$ times, the base salary.