Solve: $-2(5y-5)-3y \\leq -7y+22$ Inequality Step-by-Step

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Hey guys! Let's break down this inequality step-by-step. It might look a bit intimidating at first, but trust me, it's totally manageable. We'll go through the process together, making sure everything is crystal clear. So, grab your thinking caps, and let's dive into the world of inequalities!

Understanding Inequalities

Before we jump into the specific problem, let's quickly recap what inequalities are all about. Unlike equations that have a single solution, inequalities deal with a range of values. Think of it like this: instead of saying y equals a specific number, we're saying y is greater than, less than, greater than or equal to, or less than or equal to a certain number. These comparisons are what make inequalities so versatile in representing real-world situations where exact values aren't always the name of the game.

Why Inequalities Matter

Inequalities pop up everywhere! From figuring out how much you can spend within your budget to determining the minimum score you need to ace a test, they're incredibly practical. In math and science, they help us define ranges, set limits, and model situations where things aren't always perfectly equal. Mastering inequalities opens doors to more advanced topics and gives you a powerful tool for problem-solving in everyday life. So, understanding how to solve them is not just about acing your math class; it's about building a skill that will come in handy in countless situations.

Key Principles for Solving Inequalities

Now, let's talk about the rules of the game when solving inequalities. Most of the techniques are similar to solving equations, but there's one crucial difference: when you multiply or divide both sides by a negative number, you flip the inequality sign. This might seem a bit odd at first, but it's essential to keep the inequality true. For example, if 2 < 4, multiplying both sides by -1 gives -2 and -4. To keep the statement true, we need to flip the sign because -2 > -4. Besides this, we can add, subtract, multiply, and divide (by positive numbers) on both sides, just like with equations. The goal is the same: isolate the variable on one side to see what values make the inequality true.

Step-by-Step Solution

Okay, let's tackle the inequality at hand: $-2(5y - 5) - 3y \\leq -7y + 22$. We'll break it down into manageable steps so you can follow along easily.

1. Distribute the -2

First up, we need to get rid of those parentheses. We do this by distributing the -2 across the terms inside: $-2 * 5y = -10y$ and $-2 * -5 = +10$. So, the inequality now looks like this:

$-10y + 10 - 3y \\leq -7y + 22$

2. Combine Like Terms

Next, let's simplify each side by combining like terms. On the left side, we have $-10y$ and $-3y$, which combine to $-13y$. Our inequality now reads:

$-13y + 10 \\leq -7y + 22$

3. Move the y Terms to One Side

Our goal is to isolate y, so let's get all the y terms on one side. To do this, we can add $13y$ to both sides of the inequality. This gives us:

$10 \\leq 6y + 22$

4. Move the Constants to the Other Side

Now, let's move the constants (the numbers without y) to the other side. We can subtract 22 from both sides:

$10 - 22 \\leq 6y$

Which simplifies to:

$-12 \\leq 6y$

5. Isolate y

Almost there! To get y by itself, we need to divide both sides by 6:

$\frac{-12}{6} \\leq y$

This simplifies to:

$-2 \\leq y$

6. Rewrite in Standard Form

It's standard practice to write the inequality with the variable on the left. So, we can rewrite $-2 \\leq y$ as:

$y \\geq -2$

Therefore, the solution is A) $y \\geq -2$

Common Mistakes to Avoid

Solving inequalities is pretty straightforward once you get the hang of it, but there are a couple of common pitfalls to watch out for. Spotting these early can save you a lot of headaches!

Forgetting to Flip the Sign

The biggest mistake? Forgetting to flip the inequality sign when you multiply or divide by a negative number. This is crucial! Always double-check this step. It's like a secret trap that's easy to miss, but flipping that sign at the right time is what keeps your answer on track.

Incorrectly Distributing Negatives

Another frequent slip-up happens during distribution, especially with negative signs. Make sure you're multiplying the negative number by every term inside the parentheses. A small mistake here can throw off the whole solution. Take your time, and maybe even rewrite the expression to make the signs super clear.

Combining Terms Incorrectly

Combining like terms is a fundamental step, but it's easy to rush and make a mistake, especially when there are lots of terms. Ensure you're only combining terms that have the same variable and exponent. Double-check your addition and subtraction to avoid errors.

Misinterpreting the Solution

Once you've solved for y, make sure you understand what the inequality is telling you. Is y greater than or less than the number? It's easy to mix this up, so take a moment to really think about what the solution means in terms of possible values for y.

Practice Problems

Alright, let's flex those inequality-solving muscles with a few practice problems. Working through these will help solidify your understanding and boost your confidence. Remember, practice makes perfect, so don't be afraid to dive in and give them a try!

Practice Problem 1

Solve the inequality: $3(x + 2) > 5x - 8$

Practice Problem 2

Solve the inequality: $-4(2a - 1) \\leq -6a + 10$

Practice Problem 3

Solve the inequality: $7y - 3(y + 2) \\geq 2y - 5$

Real-World Applications

Inequalities aren't just abstract math concepts; they're super practical tools that show up in all sorts of real-world scenarios. Understanding them can help you make smarter decisions every day.

Budgeting and Finance

Think about budgeting. You have a certain amount of money to spend each month, and you want to make sure your expenses don't exceed that amount. Inequalities are perfect for this! You can set up an inequality to represent your spending limit and figure out how much you can allocate to different categories like rent, food, and entertainment.

Health and Fitness

Inequalities also play a role in health and fitness. For example, if you're trying to lose weight, you might set a goal for your daily calorie intake. You can use an inequality to represent the maximum number of calories you want to consume each day. Similarly, if you're training for a marathon, you might have a target range for your weekly mileage. Inequalities can help you stay within those limits.

Engineering and Science

In fields like engineering and science, inequalities are used to define safety margins and tolerances. For instance, when designing a bridge, engineers need to ensure that the structure can withstand a certain range of loads. Inequalities help them set those limits and ensure the bridge's safety. In chemistry, inequalities can be used to represent the range of conditions under which a reaction will occur.

Everyday Decision Making

Even in everyday situations, inequalities can guide your decisions. Imagine you're planning a road trip and want to make sure you have enough gas to reach your destination. You can use inequalities to calculate the minimum amount of gas you need based on the distance and your car's fuel efficiency.

Conclusion

So, there you have it! We've successfully navigated the world of inequalities and solved the problem at hand. Remember, the key is to take it step by step, pay close attention to those negative signs, and practice, practice, practice. Inequalities might seem tricky at first, but with a little effort, you'll be solving them like a pro in no time. Keep up the great work, and happy problem-solving!