Unlocking Math: Solving Inequalities With Ease

Hey math enthusiasts! Ever stumbled upon a problem that's not just about finding one answer, but a range of possible solutions? That's where inequalities come into play. Today, we're diving into a fun math problem: "A number divided by 3 is at most 11." Using $x$ to represent the number, we'll figure out which expression perfectly captures all the possible values for this number. Get ready to flex those math muscles and learn how to solve inequalities with confidence, it's easier than you think!

Understanding the Basics: Inequalities Demystified

Alright, before we jump into the main question, let's make sure we're all on the same page. Inequalities are like equations, but instead of an equals sign (=), they use symbols that show a relationship where one thing is not equal to another. These symbols are the secret sauce, guys! We use them to show that something is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) something else. Think of it like a seesaw; if one side is heavier, the other side goes up.

So, why are inequalities important? Well, they're super useful in the real world! Imagine you're planning a budget. You might say, "I can spend at most $50 on groceries." That "at most" is a key phrase. It means you can spend $50, or anything less than $50, but not more. Inequalities help us represent those kinds of situations mathematically. Another practical use is when we consider speed limits on roads. You must not go faster than the speed limit, otherwise, you could be fined. It is the same as the equation where a number is at most a certain value. We can utilize inequalities to describe the different situations.

Now, let's break down the language of inequalities. We have the following symbols:

• >: Greater than.
• <: Less than.
• ≥: Greater than or equal to.
• ≤: Less than or equal to.

When we solve an inequality, our goal is to isolate the variable (in our case, $x$) and find the range of values that make the inequality true. It's similar to solving equations, but there's a slight twist. When you multiply or divide both sides of an inequality by a negative number, you have to flip the inequality sign. But don't worry, we'll cover this as we get further into our problem.

Inequality Example

Let's get even more practical! Let's say you're saving up for a new game, and you need at least $30. Your inequality might look like this: $x ≥ 30$, where $x$ represents the amount of money you have. This means you can have $30, or $31, or $32, and so on. The inequality gives us a whole bunch of numbers that make the statement true. Pretty cool, right? In the next section, we'll dive right into the core question and break it down step-by-step.

Translating the Problem into Math: Inequality Expression

Okay, time to get our hands dirty with the main question. The problem states, "A number divided by 3 is at most 11." This is a classic example of an inequality problem. Our first step is to translate this statement into a mathematical expression. Remember, we are using $x$ to represent the number. Let's break it down piece by piece. First we need to represent the number. Then the number is being divided by three, this is represented as $\frac{x}{3}$. The phrase "at most" is the most important part because it's the key to the whole question. "At most" means the value can be equal to or less than something. And in this case, the "something" is 11. Therefore, "at most 11" is expressed as $\le 11$.

So, putting it all together, we get the following:

$\frac{x}{3} ≤ 11$

This inequality perfectly captures the problem's core idea. Let's see how we would get to the answer. The goal here is to isolate $x$ and figure out the range of possible values it can have. To do this, we need to undo the division by 3. The best way to do that is to multiply both sides of the inequality by 3. And yes, you must do this on both sides, whatever you do to one side you must do it to the other to keep things fair. When you multiply $\frac{x}{3}$ by 3, the 3s cancel out, and you're left with just $x$ on the left side. On the right side, 11 multiplied by 3 gives us 33. The inequality sign remains the same because we multiplied by a positive number. This is another important rule to remember, guys.

So, our next step:

$3 * \frac{x}{3} ≤ 11 * 3$

This leads to:

$x ≤ 33$

This is our final answer. It means that the number $x$ can be 33, or anything less than 33. The solution set includes all numbers from negative infinity up to and including 33. Understanding how to translate word problems like this into inequalities is a powerful skill. It helps you to not only solve math problems but also analyze and understand real-world scenarios better.

Deep Dive: Solving the Inequality

Now, let's get into the nitty-gritty of solving this inequality, as it is a crucial step. Remember, the goal is always to get the variable ($x$ in this case) all by itself on one side of the inequality. We've already taken the most important steps, but let's go over them again to make sure we've got a firm grasp of the process.

First, we started with the inequality: $\frac{x}{3} ≤ 11$. To solve this, we need to isolate the variable $x$. The operation affecting $x$ is division by 3. To undo this, we perform the inverse operation, which is multiplication. We multiply both sides of the inequality by 3. This is what you must do to keep things balanced.

So, the next step involves multiplying both sides by 3: $3 * (\frac{x}{3}) ≤ 11 * 3$. When you do the multiplication, you get $x ≤ 33$. This is the solution to the inequality, which means $x$ can be any number that is less than or equal to 33.

Let's get super clear about what this means. This result gives us a range of values that $x$ can take on. For instance, $x$ could be 33, 32, 20, 0, or even -100. The key is that the value of $x$ is never greater than 33. A key aspect of understanding inequalities is being able to visualize the solution on a number line. On a number line, you'd draw a closed circle at 33 (because 33 is included in the solution) and then shade the line to the left of 33, representing all the numbers less than 33.

Verification and Checking the Answers

Let's prove it with a few examples. Let's check a few values to make sure everything lines up:

1. Checking $x=33$: If $x = 33$, then $\frac{33}{3} = 11$. Since 11 is equal to 11, this value satisfies the original inequality $\frac{x}{3} ≤ 11$.
2. Checking $x=30$: If $x = 30$, then $\frac{30}{3} = 10$. Since 10 is less than 11, this value also works.
3. Checking $x=0$: If $x = 0$, then $\frac{0}{3} = 0$. Since 0 is less than 11, it also fits the solution set.
4. Checking $x=34$: If $x = 34$, then $\frac{34}{3} = 11.33$. This is not less than or equal to 11, so this is not part of the solution.

These examples confirm that our solution, $x ≤ 33$, is spot-on. It's a great habit to plug in different values to check if your answer makes sense. This helps you catch any mistakes and build your confidence in your math skills.

Conclusion: Mastering Inequalities

Awesome work, everyone! You've successfully navigated the world of inequalities and solved a tricky problem. We've translated a word problem, crafted a mathematical expression, solved for the variable, and even checked our answers to make sure they were correct. Remember the key takeaways:

• Understanding the symbols: Know the difference between >, <, ≥, and ≤.
• Translating the language: Learn to convert words into mathematical expressions.
• Solving for the variable: Isolate the variable using inverse operations, and always remember to do the same thing to both sides of the inequality.
• Checking your work: Plug in values to make sure your solution makes sense.

With practice, you'll become more and more comfortable with inequalities. Keep practicing, and you'll find that these kinds of math problems become much easier. And that will translate to more confidence in your math skills! Keep up the great work and happy solving! You've got this, guys! Remember that math can be fun, and with the right approach, it's easier to master than you might think.