Simple Inequality: Solve 1/3 B - 4 <= 4

by Andrew McMorgan 40 views

Hey math whizzes and welcome back to Plastik Magazine! Today, we're diving into the world of inequalities, specifically tackling a pretty straightforward one: 13bβˆ’4≀4\frac{1}{3} b-4 \leq 4. Don't let the fraction or the inequality sign scare you, guys. We're going to break this down step-by-step, making sure you understand every move. Inequalities are super important in math because they help us describe a range of possible values, not just a single answer like in equations. Think about it – sometimes you don't need an exact number, you just need to know if something is greater than, less than, or equal to a certain value. This is common in real-world scenarios, like figuring out the maximum speed you can drive without getting a ticket or the minimum amount of money you need to save for a purchase. So, let's get our hands dirty and solve this inequality together. We'll aim to isolate the variable 'bb' on one side of the inequality sign. This means we need to get rid of that '-4' and the '13\frac{1}{3}' that are hanging out with 'bb'. Remember, whatever we do to one side of the inequality, we must do to the other side to keep it balanced. It's like a seesaw – if you add weight to one side, you have to add the same weight to the other to keep it level. Our first goal is to get rid of the '-4'. How do we do that? By doing the opposite operation, which is adding 4. So, we'll add 4 to both sides of our inequality. This will cancel out the '-4' on the left side, leaving us with just the term containing 'bb'. After that, we'll tackle the fraction '13\frac{1}{3}'. Since 'bb' is being multiplied by '13\frac{1}{3}', we'll need to perform the inverse operation, which is division by '13\frac{1}{3}', or more commonly, multiplying by its reciprocal, which is 3. Again, we must do this to both sides. By the end, we'll have 'bb' all by itself, and we'll know the range of values that satisfy the original inequality. This process is fundamental to solving all sorts of algebraic problems, so pay close attention! We're going to make sure you feel confident tackling similar problems. Let's get started!

Step 1: Isolate the term with the variable

Alright team, the first move in solving our inequality, 13bβˆ’4≀4\frac{1}{3} b-4 \leq 4, is to get the term with our variable, 'bb', all by itself on one side. Right now, we have '13b\frac{1}{3} b' and then we have that '-4' chilling next to it. To start peeling away the numbers from 'bb', we always want to deal with addition and subtraction before multiplication and division. It's like unwrapping a present – you usually take off the outer layers (addition/subtraction) before you get to the ribbon and bow (multiplication/division). So, to get rid of that '-4', we're going to perform its opposite operation, which is adding 4. Crucially, remember the golden rule of inequalities (and equations, for that matter): whatever you do to one side, you must do to the other side. If we only added 4 to the left side, the whole balance would be thrown off. We want to maintain the relationship described by the '≀\leq' sign. So, let's add 4 to both the left side and the right side of our inequality:

13bβˆ’4+4≀4+4\frac{1}{3} b - 4 + 4 \leq 4 + 4

On the left side, the '-4' and '+4' cancel each other out, leaving us with just '13b\frac{1}{3} b'. On the right side, 4 plus 4 equals 8. So, our inequality simplifies to:

13b≀8\frac{1}{3} b \leq 8

See? We've already made significant progress! The term with 'bb' is now isolated. It's much cleaner, right? This step is all about basic arithmetic and understanding how to undo operations. If you had a '+4' instead of a '-4', you would subtract 4 from both sides. If you had a '+5', you'd subtract 5, and so on. Always look for the term being added or subtracted to the variable term and do the opposite to both sides. This gets us one step closer to finding out what values of 'bb' will make this statement true. Keep this simplified inequality in mind as we move to the next stage. We're building towards the final answer, and each step is a building block. You guys are doing great!

Step 2: Eliminate the coefficient

Okay, we've successfully isolated the term with our variable. We're now looking at the inequality: 13b≀8\frac{1}{3} b \leq 8. Our variable 'bb' is currently being multiplied by '13\frac{1}{3}'. To get 'bb' completely by itself, we need to undo this multiplication. The opposite of multiplying by '13\frac{1}{3}' is dividing by '13\frac{1}{3}'. However, dividing by a fraction can sometimes be a bit tricky, so a common and often easier method is to multiply by the reciprocal of the fraction. The reciprocal of '13\frac{1}{3}' is '31\frac{3}{1}', which is just 3. So, we're going to multiply both sides of our inequality by 3.

Remember, the fundamental principle remains: what you do to one side, you must do to the other. This ensures our inequality stays true. Let's multiply:

3Γ—(13b)≀3Γ—83 \times \left( \frac{1}{3} b \right) \leq 3 \times 8

On the left side, when we multiply 3 by '13\frac{1}{3}', they cancel each other out because 3Γ—13=31Γ—13=33=13 \times \frac{1}{3} = \frac{3}{1} \times \frac{1}{3} = \frac{3}{3} = 1. So, we're left with just 1b1b, which is simply 'bb'.

On the right side, we multiply 3 by 8, which gives us 24.

So, our inequality becomes:

b≀24b \leq 24

And there you have it! We've successfully solved the inequality. This result tells us that any value of 'bb' that is less than or equal to 24 will satisfy the original inequality 13bβˆ’4≀4\frac{1}{3} b-4 \leq 4. This is our final solution. We've moved 'bb' from being part of a more complex expression to being isolated, giving us a clear understanding of the possible values it can take. This process of isolating the variable by performing inverse operations on both sides is a core skill in algebra. Whether you're dealing with simple inequalities like this one or much more complex equations, the strategy is often the same: simplify, isolate, and solve. You guys crushed it! Keep practicing these steps, and you'll be a pro in no time.

Understanding the Solution: What does b≀24b \leq 24 mean?

So, we've landed on the solution b≀24b \leq 24. What does this actually mean in the grand scheme of things, guys? It's not just a random mathematical statement; it describes a whole set of numbers that make our original inequality, 13bβˆ’4≀4\frac{1}{3} b-4 \leq 4, true. The symbol '≀\leq' means 'less than or equal to'. This tells us two things about the possible values for 'bb':

  1. bb can be exactly 24: If we substitute b=24b = 24 back into the original inequality, we should get a true statement. Let's check: 13(24)βˆ’4\frac{1}{3}(24) - 4. Well, 13\frac{1}{3} of 24 is 8. So, 8βˆ’4=48 - 4 = 4. Is 4≀44 \leq 4? Yes, it is! Since 4 is equal to 4, the inequality holds true. This confirms that 24 is part of our solution set.

  2. bb can be any number less than 24: This means any number that comes before 24 on the number line is also a valid solution. For example, let's try b=21b = 21. Substituting this into the original inequality: 13(21)βˆ’4\frac{1}{3}(21) - 4. 13\frac{1}{3} of 21 is 7. So, 7βˆ’4=37 - 4 = 3. Is 3≀43 \leq 4? Absolutely! The inequality is true.

Let's try a number that is not less than or equal to 24, say b=27b = 27. Substituting this in: 13(27)βˆ’4\frac{1}{3}(27) - 4. 13\frac{1}{3} of 27 is 9. So, 9βˆ’4=59 - 4 = 5. Is 5≀45 \leq 4? No, it's not. This confirms that values greater than 24 do not satisfy the inequality.

Visualizing the Solution:

On a number line, we would represent this solution by drawing a solid circle (or a closed circle) at the number 24. This solid circle indicates that 24 is included in the solution set because of the 'equal to' part of '≀\leq'. Then, we would shade the line to the left of 24. This shading represents all the numbers less than 24 – the infinite possibilities for 'bb' that make the inequality true. This visual representation is super helpful for understanding inequalities. It clearly shows the boundary (24) and the direction of all acceptable values.

Why is this important?

Understanding the solution set of an inequality is critical in many areas of math and science. For instance, if you were calculating the maximum load a bridge could safely hold, and your calculations resulted in a variable 'L' (for load) being L≀1000L \leq 1000 tons, you'd immediately know that any load up to and including 1000 tons is safe, but anything over that is potentially dangerous. Similarly, if you're planning a budget and find that your spending 'S' must be S≀$500S \leq \$500, you know you can spend up to 500,butnomore.Inequalitiesprovidepracticallimitsandranges,makingthemincrediblyusefultoolsforproblemβˆ’solvingintherealworld.So,nexttimeyouseeaninequality,rememberitβ€²snotjustaboutfindingβ€²500, but no more. Inequalities provide practical limits and ranges, making them incredibly useful tools for problem-solving in the real world. So, next time you see an inequality, remember it's not just about finding 'b

, but about defining a whole world of possibilities!

Conclusion: Mastering Simple Inequalities

So there you have it, math enthusiasts! We've successfully navigated the inequality 13bβˆ’4≀4\frac{1}{3} b-4 \leq 4 and arrived at the solution b≀24b \leq 24. We broke it down into manageable steps: first, isolating the variable term by dealing with the constant, and second, eliminating the coefficient by multiplying by the reciprocal. Each step involved applying the fundamental rule of inequalities – whatever you do to one side, you must do to the other to maintain balance. The result, b≀24b \leq 24, means that any number less than or equal to 24 will make the original statement true. We even took a moment to visualize this on a number line and discussed its real-world implications, proving that math isn't just confined to textbooks; it's a powerful tool for understanding and managing the world around us.

Mastering simple inequalities like this is a foundational skill in algebra. It builds confidence and prepares you for more complex problems involving inequalities, systems of inequalities, and their applications in graphing and optimization. The principles are consistent: identify the variable, use inverse operations to isolate it, and always perform the same operation on both sides. Don't shy away from fractions or negative numbers; they're just part of the mathematical landscape, and with practice, they become easy to handle.

Remember, the goal is always to isolate the variable. Think of it like a puzzle where you're trying to get 'bb' alone in its own box. Every operation you perform is a move to get it closer to that goal. Keep practicing these techniques with different numbers and different operations. The more you solve, the more intuitive these steps will become. Whether you're a student grinding through homework, preparing for a test, or just a curious mind wanting to sharpen your math skills, these simple inequalities are excellent practice grounds. We hope this breakdown has been clear and helpful. Keep that mathematical curiosity alive, and we'll see you in the next article on Plastik Magazine!