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Algebra / Linear inequalities in one or two variables Difficulty: Hard

A salesperson’s total earnings consist of a base salary of $x$ dollars per year, plus commission earnings of $11 percent sign$ 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 the base salary. Which of the following inequalities represents all possible values of total sales $s$ , in dollars, the salesperson can make this year in order to meet that goal?

$2 x \leq s \leq 3 x$

$\frac{2}{0.11} x \leq s \leq \frac{3}{0.11} x$

$3 x \leq s \leq 4 x$

$\frac{3}{0.11} x \leq s \leq \frac{4}{0.11} x$

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Explanation

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 $11 percent sign$ 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 $x plus 0.11 s$ . 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 $3 x$ and $4 x$ , respectively. Therefore, this situation can be represented by the inequality $3 x \leq x + 0.11 s \leq 4 x$ . Subtracting $x$ from each part of this inequality yields $2 x \leq 0.11 s \leq 3 x$ . Dividing each part of this inequality by $0.11$ yields $\frac{2}{0.11} x \leq s \leq \frac{3}{0.11} x$ . Therefore, the inequality $\frac{2}{0.11} x \leq s \leq \frac{3}{0.11} x$ 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.