Algebra 1 word problems for 'and' compound inequalities

📚 Understanding 'And' Compound Inequalities

An 'and' compound inequality combines two inequalities with the condition that both must be true simultaneously. This means the solution is the intersection of the solutions to each individual inequality. Think of it as finding where the two inequalities overlap on a number line.

📜 Historical Context

The concept of inequalities has been around for centuries, but the formal use of compound inequalities became more prevalent with the development of symbolic algebra. Early mathematicians used geometric methods to solve problems that we now tackle with algebraic inequalities. The 'and' condition helped refine solutions to specific ranges, essential in many applications.

📌 Key Principles

• 🔍 Intersection: 'And' means the solution must satisfy both inequalities. This results in an intersection of the solution sets.
• 📈 Graphing: On a number line, the solution is the region where the graphs of the two inequalities overlap.
• ✍️ Solving: Solve each inequality separately, then find the common solution.

➕ Solving Compound Inequalities: A Step-by-Step Approach

Let's illustrate the process with an example: $2 < x + 1 \le 5$

1. Split the compound inequality: Separate the given inequality into two simpler inequalities: $2 < x + 1$ and $x + 1 \le 5$
2. Solve the first inequality:$2 < x + 1 \Rightarrow 2 - 1 < x \Rightarrow 1 < x$
3. Solve the second inequality:$x + 1 \le 5 \Rightarrow x \le 5 - 1 \Rightarrow x \le 4$
4. Combine the solutions:Since it is an 'and' compound inequality, the solution is where both conditions are met. Therefore, $1 < x \le 4$.

🌍 Real-World Examples

Here are some examples of how 'and' compound inequalities appear in real-world scenarios:

1. 🌡️ Temperature Range: A refrigerator needs to maintain a temperature between 34°F and 40°F ($34 \le T \le 40$).
2. ⏰ Time Constraint: You need to arrive at the meeting between 1:00 PM and 1:30 PM ($1 \le t \le 1.5$, where t is the time in hours past noon).
3. 💰 Budgeting: You want to spend more than $20 but no more than $50 on a gift ($20 < c \le 50$, where c is the cost).

📝 Practice Quiz

Let's test your understanding with a few word problems:

1. 🏋️‍♀️ Problem 1: A personal trainer recommends that a client's heart rate during exercise should be more than 120 beats per minute but less than or equal to 160 beats per minute. Write a compound inequality to represent the client's target heart rate (h). Answer: $120 < h \le 160$
2. 📚 Problem 2: To qualify for a certain scholarship, a student's GPA must be greater than or equal to 3.5 and their ACT score must be greater than or equal to 30. Write inequalities representing the requirements for GPA (g) and ACT score (a). Answer: $g \ge 3.5$ and $a \ge 30$
3. 🌱 Problem 3: A plant needs to be watered when the soil moisture level is below 40% but must not be watered if the level is above 70%. Write a compound inequality for the soil moisture level (m) when the plant needs water. Answer: $m < 40$
4. 🚌 Problem 4: A school bus can carry between 40 and 60 students, inclusive. Write a compound inequality representing the possible number of students (s) on the bus. Answer: $40 \le s \le 60$
5. 🎨 Problem 5: An artist wants to create a painting that is wider than 12 inches but shorter than or equal to 18 inches. Write a compound inequality for the width (w) of the painting. Answer: $12 < w \le 18$
6. 🏐 Problem 6: A volleyball team needs to have between 6 and 12 players, inclusive, to be eligible for a tournament. Write a compound inequality for the number of players (p) on the team. Answer: $6 \le p \le 12$
7. 🎢 Problem 7: To ride a certain roller coaster, a person must be taller than 48 inches but no more than 76 inches. Write a compound inequality representing the possible height (h) of a rider. Answer: $48 < h \le 76$

✅ Conclusion

Mastering 'and' compound inequalities involves understanding the concept of intersection and practicing the step-by-step solution process. By applying these principles and working through real-world examples, you can confidently tackle any 'and' compound inequality word problem!