Two-step equations and inequalities Transitional Reasoning Practice

Questions

1. 1.

   A community bicycle repair cooperative starts the month with a balance of -$150 due to purchasing a bulk shipment of spare parts. The cooperative earns a net income of$25 for each comprehensive bicycle tune-up, $t$, completed by its volunteers. To afford a new hydraulic workstand, the cooperative must bring its account balance to at least $350. Which inequality models this situation, and what is the minimum number of tune-ups that must be completed?

   • A) $25t - 150 \ge 350$; at least 20 tune-ups
   • B) $25t + 150 \ge 350$; at least 8 tune-ups
   • C) $25t - 150 \le 350$; at most 20 tune-ups
   • D) $25t + 150 \le 350$; at most 8 tune-ups
2. 2.

   A competitive archer is competing in a four-round tournament where the first three rounds have already been completed for a combined total of 258 points. To qualify for the regional championship, the archer's overall mean score across all four rounds must be at least 88 points per round. Which inequality models the score, $s$, the archer must achieve in the fourth round, and what is the required qualifying score?

   • A) $\frac{s + 258}{4} \le 88$; at most 94 points
   • B) $\frac{s + 258}{3} \ge 88$; at least 6 points
   • C) $\frac{s + 258}{4} \ge 88$; at least 88 points
   • D) $\frac{s + 258}{4} \ge 88$; at least 94 points
3. 3.

   A landscape architect is designing a rectangular community garden that must be fully enclosed by a decorative wooden border. The garden is planned to have a fixed width of 14 meters, and the architect has at most 76 meters of border material available. Which inequality models the possible length, $l$, in meters, that the garden can have, and what is the maximum possible length?

   • A) $2l + 14 \le 76$; maximum length is 31 meters
   • B) $2l + 28 \le 76$; maximum length is 24 meters
   • C) $2l + 28 \ge 76$; minimum length is 24 meters
   • D) $l + 28 \le 76$; maximum length is 48 meters
4. 4.

   A high school science club is raising funds by selling customized insulated water bottles. The supplier charges a fixed design and shipping setup fee of $48, and each bottle costs$7 to produce. If the club spends its entire allocated budget of $440 on production, which equation models the number of water bottles, $b$, the club can order, and how many bottles will they receive?

   • A) $48b + 7 = 440$; $b = 9$ bottles
   • B) $7b - 48 = 440$; $b = 69$ bottles
   • C) $7b + 48 = 440$; $b = 56$ bottles
   • D) $7(b + 48) = 440$; $b = 15$ bottles
5. 5.

   A baker divides a large batch of dough equally into 6 identical trays to make specialty loaves. Before baking, the baker removes 45 grams of excess dough from each tray, leaving each tray with exactly 380 grams of dough. Which equation models the total initial mass of the dough batch, $d$, in grams, and what was the batch's initial mass?

   • A) $\frac{d - 45}{6} = 380;\quad d = 2325\text{ g}$
   • B) $\frac{d}{6} - 45 = 380;\quad d = 2550\text{ g}$
   • C) $\frac{d}{6} + 45 = 380;\quad d = 2010\text{ g}$
   • D) $\frac{d + 45}{6} = 380;\quad d = 2235\text{ g}$
6. 6.

   An industrial cooling chamber starts at an internal temperature of $18^\circ\text{C}$. A cooling unit reduces the internal temperature at a constant rate of $4^\circ\text{C}$ per hour, and the chamber must reach a temperature of at most $-14^\circ\text{C}$ to preserve specialized biological samples. Which inequality models the number of hours, $t$, the cooling unit must operate, and what is the minimum operating time required?

   • A) $18 - 4t \le -14$, so the cooling unit must operate for at least 8 hours.
   • B) $18 - 4t \le -14$, so the cooling unit must operate for at most 8 hours.
   • C) $18 - 4t \ge -14$, so the cooling unit must operate for at least 8 hours.
   • D) $18 - 4t \le -14$, so the cooling unit must operate for at least 1 hour.
7. 7.

   A community robotics club has a maximum budget of $320 to run a workshop. The club pays a one-time room reservation fee of$65 and spends $15 on materials for each attending participant. Which inequality models the possible number of participants, $p$, the club can accommodate without exceeding its budget, and what is the maximum number of participants?

   • A) $15p + 65 \le 320$; maximum of 16 participants
   • B) $15p + 65 \le 320$; maximum of 17 participants
   • C) $15p - 65 \le 320$; maximum of 25 participants
   • D) $15p + 65 \ge 320$; minimum of 17 participants

Answers and step-by-step solutions appear after you finish the test.