The Temperature In A Laboratory Must Be Kept Within 3 Degrees Of $25^{\circ} C$. Which Of The Following Inequalities Represents This Situation?A. ∣ T + 25 ∣ ≤ 3 |T+25| \leq 3 ∣ T + 25∣ ≤ 3 B. ∣ T − N ≥ 5 ∣ ≤ 3 |T-n \geq 5| \leq 3 ∣ T − N ≥ 5∣ ≤ 3 C. ∣ T − 25 ∣ ≥ 3 |T-25| \geq 3 ∣ T − 25∣ ≥ 3 D.

Introduction

In a laboratory setting, maintaining a precise temperature is crucial for various experiments and research. The temperature must be kept within a narrow range to ensure accurate results and prevent any potential damage to equipment or samples. In this article, we will explore the mathematical representation of this temperature constraint using inequalities.

Understanding the Temperature Constraint

The temperature in the laboratory must be kept within 3 degrees of 25°C. This means that the temperature can vary between 22°C and 28°C. To represent this situation mathematically, we need to use an inequality that captures this range.

Absolute Value Inequalities

Absolute value inequalities are used to represent the distance between two values. In this case, we want to find the inequality that represents the distance between the temperature (T) and 25°C.

Option A: $|T+25| \leq 3$

This inequality represents the distance between T and -25. However, we are interested in the distance between T and 25. To fix this, we can rewrite the inequality as $|T-25| \leq 3$. This is the correct representation of the temperature constraint.

Option B: $|T-n \geq 5| \leq 3$

This inequality is incorrect because it represents the distance between T and n, where n is not defined. Additionally, the inequality is not in the correct format, as it should be $|T-n| \leq 3$.

Option C: $|T-25| \geq 3$

This inequality represents the distance between T and 25, but it is in the wrong direction. The correct inequality should be $|T-25| \leq 3$, which represents the temperature being within 3 degrees of 25°C.

Conclusion

In conclusion, the correct inequality that represents the temperature constraint in the laboratory is $|T-25| \leq 3$. This inequality captures the range of temperatures that are within 3 degrees of 25°C, ensuring that the laboratory maintains a precise temperature for various experiments and research.

Solving Absolute Value Inequalities

To solve absolute value inequalities, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: $T-25 \geq 0$

When $T-25 \geq 0$, we can rewrite the inequality as $T-25 \leq 3$. Solving for T, we get $T \leq 28$.

Case 2: $T-25 < 0$

When $T-25 < 0$, we can rewrite the inequality as $-(T-25) \leq 3$. Solving for T, we get $T \geq 22$.

Combining the Cases

Combining the two cases, we get $22 \leq T \leq 28$. This represents the range of temperatures that are within 3 degrees of 25°C.

Graphical Representation

The inequality $|T-25| \leq 3$ can be represented graphically as a closed interval on the number line. The interval represents the range of temperatures that are within 3 degrees of 25°C.

Real-World Applications

The concept of absolute value inequalities has numerous real-world applications, including:

• Temperature control: Maintaining a precise temperature in laboratories, greenhouses, and other controlled environments.
• Distance and speed: Calculating distances and speeds in various fields, such as physics, engineering, and transportation.
• Financial analysis: Analyzing financial data, such as stock prices and returns, to make informed investment decisions.

Conclusion

Q&A: Frequently Asked Questions

Q: What is the purpose of maintaining a precise temperature in a laboratory? A: Maintaining a precise temperature in a laboratory is crucial for various experiments and research. It ensures accurate results, prevents potential damage to equipment or samples, and allows for precise control over chemical reactions.

Q: How is the temperature constraint represented mathematically? A: The temperature constraint is represented mathematically using an absolute value inequality, specifically $|T-25| \leq 3$. This inequality captures the range of temperatures that are within 3 degrees of 25°C.

Q: What is the difference between $|T-25| \leq 3$ and $|T-25| \geq 3$? A: The inequality $|T-25| \leq 3$ represents the temperature being within 3 degrees of 25°C, while the inequality $|T-25| \geq 3$ represents the temperature being more than 3 degrees away from 25°C.

Q: How do you solve absolute value inequalities? A: To solve absolute value inequalities, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative. You then solve each case separately and combine the results.

Q: What are some real-world applications of absolute value inequalities? A: Absolute value inequalities have numerous real-world applications, including temperature control, distance and speed calculations, and financial analysis.

Q: How do you represent absolute value inequalities graphically? A: Absolute value inequalities can be represented graphically as a closed interval on the number line. The interval represents the range of values that satisfy the inequality.

Q: What is the significance of the number 25 in the inequality $|T-25| \leq 3$? A: The number 25 represents the target temperature, which is the temperature that the laboratory needs to maintain. The inequality $|T-25| \leq 3$ ensures that the temperature is within 3 degrees of this target temperature.

Q: Can you provide an example of how to use absolute value inequalities in a real-world scenario? A: Yes, consider a scenario where a company needs to transport a package from one location to another. The company wants to ensure that the package arrives within a certain time frame. Using absolute value inequalities, the company can calculate the maximum speed required to ensure that the package arrives on time.

Q: How do you determine the correct inequality to use in a given situation? A: To determine the correct inequality to use, you need to consider the context of the problem and the constraints that are given. In this case, the temperature constraint is given as $|T-25| \leq 3$, which represents the temperature being within 3 degrees of 25°C.

Q: Can you provide a summary of the key concepts discussed in this article? A: Yes, the key concepts discussed in this article include:

• The importance of maintaining a precise temperature in a laboratory
• The representation of temperature constraints using absolute value inequalities
• The solution of absolute value inequalities using two cases
• The graphical representation of absolute value inequalities
• Real-world applications of absolute value inequalities
• The significance of the number 25 in the inequality $|T-25| \leq 3$