Solving Inequalities: Find 's' In (s+7)/-12 ≤ 1

Hey everyone! Today, we're diving into a fun little math problem where we need to solve for s in the inequality (s+7)/-12 ≤ 1. Inequalities might seem a bit tricky at first, but don't worry, we'll break it down step-by-step so it's super easy to understand. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have an inequality, which means we're not just looking for one specific value of s. Instead, we want to find all the values of s that make the inequality true. In other words, we want to find all s such that when you add 7 to it, divide by -12, the result is less than or equal to 1.

Why Inequalities Matter

Inequalities pop up everywhere in real life. Think about speed limits on roads – you can drive at or below the limit, but not above. Or consider budgeting – you want your expenses to be less than or equal to your income. Understanding how to solve inequalities helps us make decisions and understand constraints in various situations. For example, if s represents the number of items you sell and the inequality represents your profit margin, solving for s tells you the minimum number of items you need to sell to reach a certain profit level. Understanding and solving inequalities helps in resource allocation, ensuring you stay within acceptable limits, and optimization problems where you want to find the best possible outcome within given constraints.

Key Concepts to Keep in Mind

When working with inequalities, there are a few key things to remember. First, multiplying or dividing both sides of an inequality by a negative number flips the direction of the inequality sign. This is super important and a common place where people make mistakes. For instance, if you have -x < 5, multiplying both sides by -1 gives you x > -5. Second, remember that whatever operation you perform on one side of the inequality, you must also perform on the other side to maintain the balance. Third, always double-check your solution by plugging it back into the original inequality to make sure it holds true. This helps to avoid careless mistakes and confirms that your answer is correct. Keeping these concepts in mind will help you tackle any inequality problem with confidence.

Step-by-Step Solution

Okay, let's get into the actual solving process.

Step 1: Multiply Both Sides by -12

Our inequality is (s+7)/-12 ≤ 1. To get rid of the fraction, we'll multiply both sides by -12. But remember the golden rule: since we're multiplying by a negative number, we need to flip the inequality sign.

So, we get: s + 7 ≥ -12

Step 2: Isolate s

Now, we want to get s all by itself on one side of the inequality. To do this, we'll subtract 7 from both sides:

s + 7 - 7 ≥ -12 - 7

This simplifies to: s ≥ -19

Step 3: The Solution

And that's it! Our solution is s ≥ -19. This means that any value of s that is greater than or equal to -19 will satisfy the original inequality.

Verifying the Solution

To make sure we didn't mess up (and it's always a good idea to check!), let's pick a value for s that fits our solution and plug it back into the original inequality.

Test Value: s = -19

Let's try s = -19:

(-19 + 7) / -12 ≤ 1

-12 / -12 ≤ 1

1 ≤ 1

This is true! So, s = -19 works.

Test Value: s = -18

Now let's try s = -18 (a number greater than -19):

(-18 + 7) / -12 ≤ 1

-11 / -12 ≤ 1

11/12 ≤ 1

This is also true, since 11/12 is less than 1. This confirms that our solution s ≥ -19 is correct.

Why Verification is Important

Verifying your solution is a critical step in solving inequalities and equations. It helps you catch any errors you might have made during the solving process, such as forgetting to flip the inequality sign or making a simple arithmetic mistake. By plugging your solution back into the original problem, you can ensure that your answer is accurate and reliable. This practice builds confidence in your problem-solving skills and reduces the likelihood of submitting incorrect answers, especially in exams or practical applications.

Common Mistakes to Avoid

When solving inequalities, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy.

Forgetting to Flip the Inequality Sign

The most common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Always remember this rule to ensure you get the correct solution.

Arithmetic Errors

Simple arithmetic errors can also lead to incorrect solutions. Double-check your calculations, especially when dealing with negative numbers.

Incorrectly Distributing

If the inequality involves parentheses, make sure to distribute correctly. For example, if you have 2(x + 3) < 10, distribute the 2 to both x and 3.

Not Simplifying Properly

Before solving, simplify both sides of the inequality as much as possible. This can make the problem easier to solve and reduce the chance of making mistakes.

Misinterpreting the Solution

Finally, make sure you understand what your solution means. For example, if you get x > 5, remember that this means x can be any number greater than 5, not just 5 itself.

Practical Applications

Understanding inequalities isn't just about solving math problems; it's also incredibly useful in real-world scenarios. Let's look at some practical applications where inequalities come into play.

Budgeting and Finance

In personal finance, inequalities help you manage your budget. For example, if you want to save at least $500 per month, you can write this as savings ≥ $500. Similarly, if you want to keep your expenses below a certain amount, you can use an inequality to represent that constraint.

Engineering and Construction

Engineers use inequalities to ensure structures are safe and can withstand certain loads. For instance, the maximum weight a bridge can support can be represented as an inequality: load ≤ maximum capacity.

Health and Nutrition

In health, inequalities can help you track your fitness goals. For example, if you want to burn more than 300 calories per workout, you can write this as calories burned > 300. Similarly, you can use inequalities to monitor your intake of nutrients like sugar or fat.

Business and Economics

Businesses use inequalities to analyze costs, revenue, and profits. For example, a company might want to determine the minimum number of products they need to sell to break even, which can be represented as revenue ≥ costs.

Environmental Science

Environmental scientists use inequalities to monitor pollution levels and ensure they stay within acceptable limits. For instance, the concentration of a pollutant in the air might need to be below a certain threshold to protect public health.

Conclusion

So, to wrap it up, solving the inequality (s+7)/-12 ≤ 1 involves multiplying both sides by -12 (and flipping the inequality sign!) and then isolating s. We found that s ≥ -19. Always remember to verify your solution to make sure it's correct. Inequalities are super useful in many real-life situations, so mastering them is a great skill to have. Keep practicing, and you'll become an inequality-solving pro in no time! Keep up the great work, guys! You've got this!