Solving The Inequality: 0.25y + 0.5 < 1.25

Hey guys! Today, we're going to dive into solving the inequality 0.25y + 0.5 < 1.25. Inequalities might seem a bit tricky at first, but once you understand the basic steps, they're really not that different from solving regular equations. We'll break it down step by step, so you can follow along easily and master this concept. So, grab your pencils and let's get started!

Understanding Inequalities

Before we jump into solving the specific inequality, let's quickly recap what inequalities are all about. Unlike equations, which show that two expressions are equal, inequalities show that two expressions are not equal. They use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding these symbols is crucial for interpreting and solving inequalities correctly.

Think of it this way: an equation is like a balanced scale, while an inequality is like a seesaw that's tilted to one side. Our goal is to figure out the range of values for the variable (in this case, 'y') that will keep the inequality true. This means we're not just looking for one specific answer, but rather a whole set of possible solutions. Inequalities are used everywhere, from calculating budgets to figuring out speed limits. They help us define boundaries and understand limits, making them an essential tool in mathematics and everyday life.

Step-by-Step Solution

Now, let's tackle the inequality 0.25y + 0.5 < 1.25. We'll follow a series of steps, much like solving a regular equation, but with a slight twist. Remember, the key is to isolate the variable 'y' on one side of the inequality. Let's break it down:

1. Isolate the Term with 'y'

Our first goal is to get the term with 'y' (which is 0.25y) by itself on one side of the inequality. To do this, we need to get rid of the +0.5. How do we do that? By subtracting 0.5 from both sides of the inequality! Remember, what you do to one side, you must do to the other to keep the inequality balanced.

So, we have:

0. 25y + 0.5 - 0.5 < 1.25 - 0.5

This simplifies to:

0. 25y < 0.75

Great! We've successfully isolated the term with 'y'. We're one step closer to finding our solution.

2. Solve for 'y'

Now that we have 0.25y < 0.75, we need to get 'y' completely by itself. This means we need to get rid of the 0.25 that's multiplying 'y'. How do we undo multiplication? You guessed it – division! We'll divide both sides of the inequality by 0.25.

So, we have:

(0.25y) / 0.25 < 0.75 / 0.25

This simplifies to:

y < 3

And there you have it! We've solved the inequality. Our solution is y < 3. This means that any value of 'y' that is less than 3 will make the original inequality true.

Interpreting the Solution

So, what does y < 3 actually mean? It means that the solution to our inequality isn't just one number, but a whole range of numbers. Any number less than 3 will satisfy the inequality. For example, 2, 0, -1, -5, and even 2.999 are all solutions! This is a key difference between inequalities and equations – equations usually have one specific solution, while inequalities have a range of solutions.

We can visualize this solution on a number line. To do this, we draw a number line and mark the number 3. Since our solution is less than 3 (not less than or equal to), we use an open circle at 3 to show that 3 itself is not included in the solution. Then, we shade the line to the left of 3 to represent all the numbers that are less than 3. This shaded region represents all the possible values of 'y' that make the inequality 0.25y + 0.5 < 1.25 true. Visualizing the solution on a number line can be super helpful for understanding what the inequality is telling us.

Common Mistakes to Avoid

When solving inequalities, there are a few common mistakes that students often make. Let's go over them so you can avoid these pitfalls:

Flipping the Inequality Sign

This is a big one! Remember, when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if you have -2y < 4, and you divide both sides by -2, you need to change the inequality sign from < to >. So, the correct solution would be y > -2. Forgetting to flip the sign is a very common mistake, so always double-check when you're working with negative numbers.

Incorrectly Distributing

If the inequality involves parentheses, you need to distribute correctly. Make sure you multiply the number outside the parentheses by every term inside the parentheses. For example, if you have 2(y + 1) < 6, you need to distribute the 2 to both 'y' and 1, resulting in 2y + 2 < 6. A simple mistake in distribution can throw off your entire solution, so take your time and be careful.

Forgetting to Perform the Same Operation on Both Sides

Just like with equations, you must perform the same operation on both sides of the inequality to keep it balanced. If you add 3 to one side, you need to add 3 to the other side. If you divide one side by 2, you need to divide the other side by 2. Forgetting this fundamental rule will lead to an incorrect solution. Think of it like a seesaw – if you only change one side, it won't be balanced anymore!

Not Checking Your Solution

Finally, it's always a good idea to check your solution by plugging it back into the original inequality. Choose a value within your solution range and see if it makes the inequality true. For example, if you found y < 3, you could plug in y = 2. If the inequality holds true, you're on the right track! If not, it's a sign that you might have made a mistake somewhere along the way.

Real-World Applications

Inequalities aren't just abstract math concepts; they have tons of real-world applications. Let's look at a few examples to see how inequalities are used in everyday life:

Budgeting

Imagine you have a budget of $50 for groceries. You want to buy some fruits and vegetables, but you need to make sure you don't spend more than $50. This situation can be represented by an inequality. If 'x' represents the amount you spend on fruits and vegetables, then the inequality would be x ≤ 50. This inequality tells you that the amount you spend ('x') must be less than or equal to $50.

Speed Limits

Speed limits are another great example of inequalities in action. A speed limit sign might say 65 mph, which means the speed you're traveling ('s') must be less than or equal to 65 mph. This can be written as the inequality s ≤ 65. Inequalities help keep our roads safe by setting limits on how fast we can drive.

Height Restrictions

Ever been to an amusement park and seen a height restriction for a ride? These restrictions are also based on inequalities. For example, a ride might have a minimum height requirement of 48 inches. If 'h' represents your height, then the inequality would be h ≥ 48. This means that your height ('h') must be greater than or equal to 48 inches to ride the ride. Inequalities ensure safety by setting limits based on physical characteristics.

Sales and Discounts

Inequalities are also used in sales and discounts. For example, a store might offer a 20% discount on items that cost more than $50. If 'c' represents the original cost of the item, then the inequality would be c > 50. This inequality tells you that you only get the discount if the original cost of the item is greater than $50.

Practice Problems

Okay, now it's your turn to practice! Here are a few problems for you to try. Remember the steps we discussed, and don't be afraid to make mistakes – that's how we learn! Give these a shot, and then check your answers.

1. Solve the inequality: 2y - 3 > 7
2. Solve the inequality: -3y + 5 ≤ 14
3. Solve the inequality: 0.5y + 2 < 4

(Answers: 1. y > 5, 2. y ≥ -3, 3. y < 4)

Conclusion

So, guys, we've covered a lot about solving the inequality 0.25y + 0.5 < 1.25! We've broken down the steps, talked about common mistakes, and even explored some real-world applications. Remember, the key to mastering inequalities is practice, practice, practice! The more you work with them, the more comfortable you'll become. Don't be afraid to ask questions and seek help when you need it. Math can be challenging, but it's also super rewarding when you finally get it. Keep up the great work, and you'll be solving inequalities like a pro in no time!