Solving Inequalities: Find Intervals For 2b > -3b + 5 - 72 - 14b

Hey guys! Let's dive into the world of inequalities today. We're going to break down how to solve the inequality 2b > -3b + 5 - 72 - 14b step by step. This might seem a bit daunting at first, but trust me, it's totally manageable once you understand the process. We'll go through each step in detail, so you'll be a pro at solving these types of problems in no time! So, grab your pencils and notebooks, and let's get started!

Understanding Inequalities

Before we jump into solving this specific problem, it’s important to understand what inequalities are and how they differ from equations. Think of an equation as a balanced scale; both sides are equal. An inequality, on the other hand, is like a scale that's tipping to one side. It shows a relationship where one side is either greater than, less than, greater than or equal to, or less than or equal to the other side.

In mathematical terms, instead of using the equals sign (=), we use symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). These symbols open up a range of possible solutions rather than just one specific value. For example, x > 5 means that x can be any number greater than 5, not just 5 itself.

When dealing with inequalities, the goal is similar to solving equations: to isolate the variable. However, there are a few key differences. One crucial rule to remember is that when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the values and their relative positions on the number line.

Understanding these basics is crucial because it forms the foundation for solving more complex inequalities, like the one we're tackling today. By grasping the concept of inequalities and how they behave, you'll be better equipped to manipulate and solve them accurately. So, let’s keep these principles in mind as we move forward and start cracking this problem!

Step-by-Step Solution

Now, let's get our hands dirty and solve the inequality 2b > -3b + 5 - 72 - 14b. We’ll take it one step at a time, just like building a house. First, we need a solid foundation, and that means simplifying the expression. Think of it as decluttering – we want to make things as neat and tidy as possible before we start rearranging.

1. Simplify the Inequality

Our first task is to combine like terms on the right side of the inequality. We have -3b and -14b, which are both terms involving the variable 'b'. Adding these together gives us -17b. We also have the constants 5 and -72. Combining these gives us -67. So, the inequality now looks like this:

2b > -17b - 67

See? We've already made progress! It looks much cleaner than it did before. Simplifying expressions like this is super important because it reduces the chances of making mistakes later on. Think of it as prepping your ingredients before you start cooking – it makes the whole process smoother and more enjoyable.

2. Isolate the Variable

The next step is to get all the terms with 'b' on one side of the inequality. This is similar to gathering all your tools in one place so you can easily reach them. To do this, we need to add 17b to both sides of the inequality. Remember, whatever we do to one side, we have to do to the other to keep things balanced. This gives us:

2b + 17b > -17b - 67 + 17b

This simplifies to:

19b > -67

We're getting closer! Now we have all the 'b' terms on the left side, which is exactly what we want. Isolating the variable is a key strategy in solving inequalities, just like focusing on the main goal in any project. By keeping the 'b' terms together, we’re setting ourselves up for the final step.

3. Solve for 'b'

Now for the final move! To solve for 'b', we need to get 'b' all by itself. It’s like untangling the last knot in a string. Since 'b' is being multiplied by 19, we need to divide both sides of the inequality by 19. This will undo the multiplication and leave us with 'b' on its own:

19b / 19 > -67 / 19

This simplifies to:

b > -67/19

And there you have it! We've solved for 'b'. This tells us that 'b' is greater than -67/19. This is our solution, and it represents a range of values that 'b' can take.

Expressing the Solution

Now that we've found the solution, let's talk about how to express it in different ways. Understanding this is like knowing how to present your finished masterpiece – you want it to look its best!

1. Interval Notation

Interval notation is a concise way to represent a range of numbers. It uses parentheses and brackets to indicate whether the endpoints are included in the interval. Since our solution is b > -67/19, we're dealing with all numbers greater than -67/19, but not including -67/19 itself. This is where parentheses come in handy.

So, in interval notation, our solution is written as:

(-67/19, ∞)

The parenthesis next to -67/19 means that -67/19 is not included in the interval. The infinity symbol (∞) represents positive infinity, and we always use a parenthesis next to infinity because infinity is not a specific number and therefore cannot be included in the interval. This notation neatly captures the idea that 'b' can be any number greater than -67/19, all the way up to infinity.

2. Number Line Representation

Another visual way to represent our solution is using a number line. This is like drawing a map to show the range of values that satisfy the inequality. To do this, draw a number line and locate the point -67/19. Since 'b' is greater than -67/19, we'll use an open circle (or a parenthesis) at this point to indicate that it's not included in the solution.

Then, we'll draw an arrow extending to the right from -67/19. This arrow represents all the numbers greater than -67/19. Shading the line along the arrow helps to visually emphasize the range of solutions. This visual representation makes it super clear which values of 'b' satisfy the inequality.

3. Set Notation

Set notation is a more formal way of expressing the solution using set theory. It involves defining a set of all values that satisfy the inequality. Our solution can be written in set notation as:

{b | b > -67/19}

This is read as “the set of all 'b' such that 'b' is greater than -67/19.” The vertical bar (|) is read as “such that.” This notation is precise and leaves no room for ambiguity. It’s like giving a detailed description of the solution, ensuring everyone understands exactly which values are included.

Common Mistakes to Avoid

When solving inequalities, there are a few common pitfalls that can trip you up if you're not careful. Let's talk about these so you can steer clear of them!

1. Forgetting to Flip the Sign

This is the big one! As we mentioned earlier, whenever you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. If you forget to do this, you'll end up with the wrong solution. It’s like driving on the wrong side of the road – it will definitely lead to a crash!

For example, if you have -2b > 4, you need to divide both sides by -2. This means you should flip the > sign to <, giving you b < -2. Always double-check this step when you see a negative number involved.

2. Incorrectly Combining Like Terms

Another common mistake is messing up the combination of like terms. This usually happens when signs get mixed up or terms are overlooked. It's like adding the wrong ingredients to a recipe – it can throw off the whole dish. Before you start moving things around in the inequality, take a moment to simplify each side separately. Make sure you've added and subtracted the correct terms and that you haven't missed anything.

3. Misinterpreting the Solution

Once you've solved for the variable, it’s crucial to understand what your solution actually means. For example, if you get b > 3, that means b can be any number greater than 3, not just 3 itself. It’s like reading a map – you need to know how to interpret the symbols to reach your destination. Make sure you understand whether your solution includes the endpoint or not, and express your answer correctly using interval notation or a number line.

4. Not Distributing Negatives Properly

If you have a negative sign in front of a parenthesis, remember to distribute it to every term inside the parenthesis. Forgetting to do this is like forgetting to close a door – it can lead to all sorts of problems. For example, if you have -(x + 2) > 5, you need to distribute the negative sign to both x and 2, giving you -x - 2 > 5. Neglecting this step can completely change the inequality and lead to an incorrect solution.

Practice Makes Perfect

Alright, guys! We've covered a lot today, from understanding inequalities to solving them step by step, and even how to express the solutions in different ways. But remember, like any skill, mastering inequalities takes practice. It’s like learning to ride a bike – you might wobble a bit at first, but with consistent effort, you’ll be cruising along smoothly in no time!

The best way to get comfortable with solving inequalities is to tackle a variety of problems. Start with simpler ones and gradually work your way up to more complex scenarios. Each problem you solve will help solidify your understanding and build your confidence. Don't be afraid to make mistakes – they're a natural part of the learning process. The key is to learn from them and keep going.

So, grab some practice problems, put on your thinking caps, and get to work! You've got this! And remember, if you ever get stuck, don't hesitate to review the steps we've discussed or seek help from a teacher, tutor, or online resources. Happy solving!