Inequalities are mathematical expressions that show the relationship between two values, indicating that one is less than, greater than, or not equal to the other. Solving inequalities is essential in various real-life applications, from budgeting to understanding limits in science. In this article, we’ll walk through three practical examples of solving inequalities, breaking them down step-by-step to ensure clarity and understanding.
Imagine you’re planning a birthday party and have a budget. You want to know how many guests you can invite without exceeding your budget.
You have $200 to spend, and each guest costs $15 for food and drinks. How many guests can you invite while staying within your budget?
To find out, we can set up the inequality:
15x ≤ 200
Where x represents the number of guests.
To solve this inequality, we divide both sides by 15:
x ≤ 200 / 15
x ≤ 13.33
Since you can’t invite a fraction of a guest, you can invite a maximum of 13 guests.
This inequality not only helps you plan your party but also teaches you to balance costs against your budget. If you wanted to invite more guests, you might consider ways to reduce costs or increase your budget.
Suppose you’re working in a lab where a chemical reaction needs to occur at a temperature above 50 degrees Celsius but below 75 degrees Celsius for optimal results. You need to express this temperature range as an inequality.
To represent this situation, you can write:
50 < T < 75
Where T represents the temperature. This compound inequality shows that the temperature must be greater than 50 degrees and less than 75 degrees.
To solve or analyze this inequality, you need to consider both parts:
This example illustrates how inequalities can be used in scientific contexts to ensure conditions are met for safe and effective reactions. Variations could involve adjusting the temperature limits based on new research findings.
You’re a runner training for a marathon and you want to maintain a certain pace to ensure you finish within a specific time. You aim to complete the marathon (which is 26.2 miles) in less than 4 hours. You want to calculate the maximum pace you can run per mile.
To set this up, we can use the inequality:
26.2 / t < 4
Where t represents the total time in hours. To solve for t:
26.2 < 4t
Now, divide both sides by 4:
t > 26.2 / 4
t > 6.55
This means you need to run faster than 6.55 miles per hour to finish the marathon in under 4 hours.
This example not only helps you plan your training but also gives insight into pacing strategies for long-distance running. You could adjust the inequality if you want to increase your finish time or distance.
By understanding and applying these examples of solving inequalities, you can tackle real-world problems with confidence and clarity.