Algebra Inequality: Find The Number Greater Than 5

Hey guys! Ever get stuck on those tricky algebra problems where you need to figure out a number based on a description? Today, we're diving into one of those, and trust me, it's not as scary as it sounds. We're going to break down an inequality problem step-by-step, so you can totally nail it. The question is: If three times a number plus four is no more than four times the number minus one, which of the following are true? Let $z$ represent the number. We've got some options to check: $z > 5$, $z less 5$, $z less 5$, and 'The number is at least 5.' Sounds like a mouthful, right? But don't sweat it! We'll translate this word problem into a mathematical expression and solve for $z$. By the end, you'll be able to confidently tackle similar problems and impress your friends with your math skills. So grab your notebooks, settle in, and let's get this math party started!

Understanding the Inequality

Alright, let's get real with this problem. The first step to solving any word problem, especially in math, is to translate the words into symbols. Think of it like learning a secret code! Here, the problem gives us a scenario: 'three times a number plus four is no more than four times the number minus one.' Let's break that down. We're told to use the variable $z$ to represent our mystery number. So, 'three times a number' becomes $3z$. Then, 'plus four' is simply $+4$. Putting that together, we get $3z + 4$. Now for the other side of the coin: 'four times the number' is $4z$, and 'minus one' is $-1$. So that part is $4z - 1$. The crucial phrase here is 'no more than.' In math terms, 'no more than' means less than or equal to. So, our inequality is $3z + 4 less 4z - 1$. This inequality is the heart of the problem, guys, and once we have it set up correctly, the rest is pretty smooth sailing. It's all about careful reading and knowing your inequality symbols. Remember, 'no more than' is your cue for $less$. Other common phrases to watch out for include 'at least' (which means $less$), 'greater than' ($>$), and 'less than' ($<$). Mastering this translation is key to unlocking the solution and making sure we're working with the right mathematical representation of the problem. So, take a moment to really digest this: $3z + 4 less 4z - 1$. This single line holds all the information we need to find out what we can about our number $z$. It's pretty cool how math can condense complex ideas into simple expressions, isn't it? Now, let's get to the fun part: solving it!

Solving for z

Now that we've translated the problem into a solid inequality, $3z + 4 less 4z - 1$, it's time to roll up our sleeves and solve for $z$. Our goal here is to isolate $z$ on one side of the inequality sign. Think of it like trying to get all the $z$'s together and all the numbers together. We can do this using the same rules we use for solving equations, with one tiny but important exception: if we multiply or divide by a negative number, we have to flip the inequality sign. But let's see if we even need to worry about that here!

First, let's get all the $z$ terms on one side. I like to keep my $z$ terms positive if I can, so I'll subtract $3z$ from both sides. Remember, whatever you do to one side, you must do to the other to keep the inequality balanced.

$3z + 4 - 3z less 4z - 1 - 3z$

This simplifies to:

$4 less z - 1$

See? That wasn't so bad. Now, we want to get $z$ all by itself. The $z$ term currently has a $-1$ with it. To undo that, we'll add 1 to both sides of the inequality.

$4 + 1 less z - 1 + 1$

And that leaves us with:

$5 less z$

So, we've found our solution: $5 less z$. What does this mean in plain English? It means that $z$ is greater than 5. We successfully isolated $z$ and determined the range of possible values for our number. This step is all about performing inverse operations to peel away the numbers and operations surrounding $z$ until $z$ is standing alone. Each step we take maintains the truth of the inequality, ensuring our final result is accurate. It's a bit like solving a puzzle, where each move brings you closer to the final picture. And in this case, the picture is that our number $z$ has to be bigger than 5. Pretty neat, right?

Interpreting the Solution and Checking the Options

Okay, math whizzes, we've done the heavy lifting and solved the inequality! We found that $5 less z$, which means $z$ must be greater than 5. Now comes the part where we connect our mathematical answer back to the original question and figure out which of the given options are true. This is where we make sure our hard work actually answers the prompt!

Let's look at our options:

• □ $z > 5$: Does our solution $5 less z$ support this? Absolutely! If $z$ is greater than 5, then $5$ is less than $z$. This option is definitely true.
• □ $z less 5$: Does our solution $5 less z$ support this? No. Our solution tells us $z$ must be greater than 5, not less than 5.
• □ $z less 5$: Let's re-read this one carefully. Oh, wait, this is the same as the previous option, $z < 5$. Again, our solution $5 less z$ contradicts this. This option is false.
• □ The number is at least 5: What does 'at least 5' mean in math terms? It means the number is greater than or equal to 5. So, this option is $z less 5$. Does our solution $5 less z$ support this? Not quite. Our solution says $z$ must be strictly greater than 5. It cannot be equal to 5. Therefore, this option is false.

So, to recap, the only statement that is definitively true based on our inequality $5 less z$ is that $z > 5$. It's super important to pay attention to the difference between 'greater than' ($>$) and 'greater than or equal to' ($less$) – that tiny symbol makes a world of difference!

Why is $z less 5$ false? Because our solution is $5 less z$. If $z$ were equal to 5, the original inequality would become $3(5) + 4 less 4(5) - 1$, which simplifies to $15 + 4 less 20 - 1$, or $19 less 19$. Since 19 is not less than or equal to 19 (it's equal), $z=5$ does not satisfy the condition. It must be strictly greater than 5. This is why 'the number is at least 5' ($z less 5$) is also false.

Conclusion: Based on our algebraic manipulation and interpretation, the only true statement among the choices is that $z > 5$. It's awesome how solving a single inequality can help us evaluate multiple potential truths. Keep practicing, and you'll become a pro at spotting these details!

Final Thoughts and Practice Tips

So there you have it, guys! We took a word problem, translated it into an algebraic inequality ($3z + 4 less 4z - 1$), solved it to find $z > 5$, and then used that solution to determine which of the given statements were true. We found that only $z > 5$ is definitely true. It’s all about breaking down the problem into manageable steps: understand, translate, solve, and interpret. This approach works wonders not just for math problems, but for tackling challenges in everyday life too!

Remember the key takeaways:

• Translate carefully: Phrases like 'no more than' ($less$), 'at least' ($less$), 'less than' ($<$), and 'greater than' ($>$) have specific mathematical meanings.
• Solve with care: Follow the rules of inequalities, especially when multiplying or dividing by negative numbers (though we didn't need to here).
• Interpret accurately: Connect your solution back to the original question and check each option thoroughly, paying close attention to strict inequalities ($>$ or $<$) versus non-strict ones ($less$ or $less$).

Want to get even better? Practice, practice, practice! Grab any inequality problems you can find – from your textbook, online resources, or even make up your own! Try to explain the steps to a friend or family member; teaching is one of the best ways to solidify your own understanding. Look for other problems that involve translating words into inequalities. For example, try problems like: 'The sum of twice a number and 7 is greater than 15,' or 'If 5 less than a number is no more than 10, what can you say about the number?' The more you work with these types of problems, the more comfortable and confident you'll become. Keep that math brain sharp, and you'll be solving complex problems like this one in your sleep! Happy solving!