Solve $3x - 3^2 \geq \sqrt{144}$: A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun little math problem that involves solving an inequality. Inequalities might seem a bit tricky at first, but trust me, they're totally manageable once you break them down step by step. We're going to tackle the inequality $3x - 3^2 \geq \sqrt{144}$. This problem combines a few different concepts, so it's a great way to brush up on your algebra skills. We’ll be covering everything from simplifying exponents and square roots to isolating the variable. So, grab your pencils, and let's get started!

Understanding the Basics

Before we jump right into solving, let’s make sure we're all on the same page with the basics. An inequality is a mathematical statement that compares two expressions using symbols like greater than ($>$), less than ($<$), greater than or equal to ($\geq$), or less than or equal to ($\leq$). Unlike an equation, which states that two expressions are equal, an inequality shows a range of possible values. Our goal here is to find all the values of $x$ that make the inequality $3x - 3^2 \geq \sqrt{144}$ true.

We also need to remember a couple of key concepts: exponents and square roots. An exponent tells us how many times to multiply a number by itself. For example, $3^2$ means $3$ multiplied by itself, which is $3 \times 3 = 9$. A square root, on the other hand, is a value that, when multiplied by itself, gives us the original number. The square root of $144$, written as $\sqrt{144}$, is the number that, when multiplied by itself, equals $144$. If you're thinking $12$, you're spot on because $12 \times 12 = 144$. So, $\sqrt{144} = 12$.

With these basics in mind, we can confidently start solving our inequality. Remember, the key to solving any math problem is to take it one step at a time and break it down into smaller, more manageable parts. Let’s get to it!

Step-by-Step Solution

Alright, let's break down this inequality step by step. Our inequality is $3x - 3^2 \geq \sqrt{144}$.

1. Simplify the Exponent and Square Root

The first thing we want to do is simplify any exponents and square roots in the inequality. This will make the problem easier to work with. We know that $3^2$ means $3$ multiplied by itself, which is $3 \times 3 = 9$. And we also know that $\sqrt{144}$ is $12$, because $12 \times 12 = 144$. So, let's replace these values in our inequality:

$3x - 9 \geq 12$

See? It already looks a bit simpler! Simplifying these terms is crucial because it reduces the complexity and allows us to focus on isolating the variable $x$.

2. Isolate the Term with $x$

Next up, we want to isolate the term with $x$ on one side of the inequality. In this case, that term is $3x$. To do this, we need to get rid of the $-9$ on the left side. We can do this by adding $9$ to both sides of the inequality. Remember, whatever we do to one side, we have to do to the other to keep the inequality balanced. This is a fundamental principle in solving equations and inequalities.

So, let's add $9$ to both sides:

$3x - 9 + 9 \geq 12 + 9$

This simplifies to:

$3x \geq 21$

Great! We're one step closer to solving for $x$. Isolating the term with the variable is a critical step because it sets us up to solve for the variable itself. By adding $9$ to both sides, we've effectively moved the constant term to the right side of the inequality.

3. Solve for $x$

Now, we need to get $x$ all by itself. Right now, $x$ is being multiplied by $3$. To undo this multiplication, we'll divide both sides of the inequality by $3$. Again, we have to do this to both sides to maintain the balance. Dividing both sides by the same positive number preserves the direction of the inequality, which is important to remember.

So, let's divide both sides by $3$:

$\frac{3x}{3} \geq \frac{21}{3}$

This simplifies to:

$x \geq 7$

And there you have it! We've solved for $x$. Our solution is $x \geq 7$, which means that $x$ can be any number that is greater than or equal to $7$.

Understanding the Solution

So, we've arrived at the solution: $x \geq 7$. But what does this actually mean? It's super important to understand the solution in the context of the problem. The inequality $x \geq 7$ tells us that any value of $x$ that is greater than or equal to $7$ will satisfy the original inequality $3x - 3^2 \geq \sqrt{144}$.

Think of it like this: if we plug in a number like $7$ for $x$, the inequality should hold true. Let's check:

$3(7) - 3^2 \geq \sqrt{144}$ becomes $21 - 9 \geq 12$, which simplifies to $12 \geq 12$. This is true!

Now, let's try a number greater than $7$, say $8$:

$3(8) - 3^2 \geq \sqrt{144}$ becomes $24 - 9 \geq 12$, which simplifies to $15 \geq 12$. This is also true!

If we were to try a number less than $7$, like $6$, we would see that the inequality does not hold:

$3(6) - 3^2 \geq \sqrt{144}$ becomes $18 - 9 \geq 12$, which simplifies to $9 \geq 12$. This is false.

This confirms that our solution $x \geq 7$ is correct. Understanding how to verify your solution is a critical skill in mathematics. It helps you ensure that your answer is correct and gives you confidence in your problem-solving abilities.

Common Mistakes to Avoid

When solving inequalities, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct solution. Let's go over some of the most frequent errors.

1. Forgetting to Flip the Inequality Sign

One of the most crucial rules to remember is that 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 $-2x > 4$, and you divide both sides by $-2$, you need to change the $>$ to $<$, resulting in $x < -2$. Forgetting this rule is a very common mistake and can lead to an incorrect solution. In our problem, we didn't encounter this situation because we only divided by positive numbers, but it's super important to keep in mind for other problems.

2. Incorrectly Simplifying Exponents and Square Roots

Another common mistake is messing up the simplification of exponents and square roots. Remember that $3^2$ means $3 \times 3$, not $3 \times 2$. Similarly, make sure you correctly identify the square root. Forgetting that $\sqrt{144}$ is $12$ and not some other number can throw off your entire solution. Always double-check these simplifications to make sure you're starting with the correct values.

3. Not Maintaining Balance

Just like with equations, it’s essential to maintain balance in inequalities. Whatever operation you perform on one side, you must perform on the other side as well. If you add a number to one side but forget to add it to the other, your inequality will become unbalanced, and you'll end up with the wrong answer. Think of it like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level.

4. Misinterpreting the Solution

Finally, make sure you understand what your solution actually means. In our case, $x \geq 7$ means that $x$ can be any number greater than or equal to $7$. It’s not just the number $7$, but a whole range of values. Misinterpreting the solution can lead to incorrect conclusions or misunderstandings of the problem.

By being mindful of these common mistakes, you can significantly improve your accuracy when solving inequalities. Remember to take your time, double-check your work, and think through each step carefully.

Practice Problems

To really nail down your understanding of solving inequalities, practice is key! Let’s try a couple of similar problems to reinforce what we’ve learned. Working through these examples will help you build confidence and become more comfortable with the process.

Practice Problem 1

Solve the inequality: $2x + 5 < 11$

Solution:

1. Subtract 5 from both sides: $2x + 5 - 5 < 11 - 5$, which simplifies to $2x < 6$.
2. Divide both sides by 2: $\frac{2x}{2} < \frac{6}{2}$, which simplifies to $x < 3$.

So, the solution is $x < 3$.

Practice Problem 2

Solve the inequality: $4x - 2 \geq 14$

Solution:

1. Add 2 to both sides: $4x - 2 + 2 \geq 14 + 2$, which simplifies to $4x \geq 16$.
2. Divide both sides by 4: $\frac{4x}{4} \geq \frac{16}{4}$, which simplifies to $x \geq 4$.

So, the solution is $x \geq 4$.

Practice Problem 3

Solve the inequality: $-3x + 1 \leq 10$

Solution:

1. Subtract 1 from both sides: $-3x + 1 - 1 \leq 10 - 1$, which simplifies to $-3x \leq 9$.
2. Divide both sides by -3 (and remember to flip the inequality sign!): $\frac{-3x}{-3} \geq \frac{9}{-3}$, which simplifies to $x \geq -3$.

So, the solution is $x \geq -3$.

Working through these practice problems gives you a chance to apply the steps we discussed earlier. Pay close attention to each step, and don’t forget about the common mistakes we talked about, like flipping the inequality sign when dividing by a negative number. The more you practice, the more natural these steps will become.

Conclusion

Alright, guys, we've made it to the end! Today, we tackled the inequality $3x - 3^2 \geq \sqrt{144}$ and walked through the entire solution process step by step. We started by simplifying the exponents and square roots, then isolated the term with $x$, and finally solved for $x$. Our solution was $x \geq 7$, which means any value of $x$ greater than or equal to $7$ will satisfy the original inequality.

We also discussed the importance of understanding the solution and how to verify it by plugging in values. Additionally, we covered some common mistakes to avoid when solving inequalities, like forgetting to flip the inequality sign when dividing by a negative number, incorrectly simplifying exponents and square roots, not maintaining balance, and misinterpreting the solution.

Remember, math is all about practice. The more you practice, the more comfortable and confident you’ll become. So, keep practicing, keep asking questions, and keep challenging yourself. You’ve got this! Solving inequalities is a fundamental skill in algebra, and mastering it will open the door to more complex mathematical concepts. Keep up the great work, and I’ll catch you in the next math adventure!