Final answer:
Hailey needs between 3 to 5 more practice sessions to meet her weekly goal of 120 to 210 minutes of practice, considering she already completed a 30-minute session. However, with just 3 sessions, she would achieve 135 minutes of practice, which meets the goal.
Step-by-step explanation:
Hailey plans to spend between 120 and 210 minutes inclusive on practicing soccer each week. Since Hailey has already practiced for 30 minutes this week, and her practice sessions are 45 minutes each, we can define a variable x as the number of additional practice sessions she needs to have. To meet her goal, the total time spent practicing must satisfy the inequality 30 + 45x ≥ 120 and 30 + 45x ≤ 210.
First, solve the inequality 30 + 45x ≥ 120 which simplifies to 45x ≥ 90, so x ≥ 2. Now solve 30 + 45x ≤ 210, which simplifies to 45x ≤ 180, so x ≤ 4. Combining both inequalities gives us 2 ≤ x ≤ 4. Because x represents additional sessions and Hailey has already done one session (of 30 minutes), we add one to each boundary, yielding 3 ≤ x+1 ≤ 5. Therefore, Hailey needs to have between 3 to 5 more practice sessions this week. However, since Hailey's practice sessions are 45 minutes each, and she has already practiced for 30 minutes, she can indeed add two more full sessions of 45 minutes to reach exactly 120 minutes(30 + 2*45 = 120), but that is the minimum boundary. To reach just over the minimum, she needs at least one more full session, making it 3 practice sessions in total for the week. If she stops at 3 sessions, that's 135 minutes which is above the minimum.