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Mastering Inequalities: Solve 5(x+27) ≥ 6(x+26) Easily

Alex Johnson
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Mastering Inequalities: Solve 5(x+27) ≥ 6(x+26) Easily

≥ 6(x+26) Easily

Hey there, math explorers! Ever looked at a math problem and thought, "Woah, what even is this?" Especially when it comes to inequalities? You're definitely not alone! These tricky expressions, which use symbols like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤), might seem intimidating at first glance. But guess what? They're super common in real life, from figuring out your budget for the month (you want to spend less than or equal to your income, right?) to understanding speed limits (you should drive less than or equal to 60 mph!). In this friendly guide, we're going to dive deep into a specific inequality problem that often stumps folks: "Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26." Don't let the wordiness scare you off! We'll break it down piece by piece, explain why each step is important, and make sure you walk away feeling like an inequality superstar. Understanding how to solve inequalities isn't just about getting the right answer on a test; it's about sharpening your logical thinking and problem-solving skills, which are useful in every aspect of life. So, grab a comfy seat, maybe a snack, and let's unravel the mysteries of inequalities together!

What Are Inequalities, Really? A Friendly Introduction

Inequalities, at their core, are simply mathematical statements that show a relationship between two expressions that are not necessarily equal. Think of them as scales that aren't perfectly balanced. While an equation tells us that two things are exactly the same (like 2 + 3 = 5), an inequality tells us that one thing is either bigger, smaller, or possibly bigger/smaller than or equal to another. It's like saying, "My spending should be less than or equal to my savings," or "The car's speed must be greater than 0." These concepts pop up everywhere! The common inequality symbols you'll encounter are: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Understanding these symbols is your very first step towards mastering inequalities. They allow us to describe ranges of values rather than a single fixed value. For example, if you need to be at least 18 years old to vote, your age, let's call it A, must satisfy A ≥ 18. This isn't a specific age like A = 18, but a whole set of possible ages (18, 19, 20, and so on, forever!). This flexibility is what makes solving inequalities so powerful for describing real-world scenarios and constraints. Our problem, "Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26," uses the ≥ symbol, indicating that one side could be larger than or exactly equal to the other. Learning to translate these word problems into proper algebraic notation is a fundamental skill that empowers you to solve a wide range of mathematical problems, from simple comparisons to complex scientific models. We'll explore how to turn this mouthful of a sentence into a clean, solvable mathematical expression in the next section, laying the groundwork for finding its solution set and truly understanding what it means.

Deconstructing Our Problem: Five Times the Sum

Let's tackle the very heart of our problem: "Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26." This sentence might look like a mouthful, but we can break it down into manageable algebraic expressions. The first crucial step in solving inequalities (or any algebra problem!) is to accurately translate the words into mathematical symbols. Whenever you see "a number" in algebra, that's your cue to introduce a variable. Let's call our mysterious number 'x'. This 'x' represents any value that could potentially make our inequality true. Now, let's take it bit by bit.

First part: "the sum of a number and 27." If our number is 'x', its sum with 27 is simply x + 27. Easy, right? Next: "Five times the sum of a number and 27." When you see "times" in mathematics, it means multiplication. So, five times (x + 27) becomes 5(x + 27). Remember to use parentheses here, because the '5' is multiplying the entire sum, not just 'x'. If we wrote 5x + 27, it would mean five times the number, then add 27, which is a very different thing! Precision in algebraic expressions is key.

Moving on to the other side of our comparison: "the sum of that number and 26." Since "that number" refers to the same 'x' we used before, this sum is x + 26. Following the same logic as before, "six times the sum of that number and 26" translates to 6(x + 26). Again, those parentheses are vital to ensure the '6' multiplies the entire sum of 'x' and '26'.

Finally, let's look at the relationship between these two parts: "is greater than or equal to." This is where our inequality symbol comes in! As we learned earlier, "greater than or equal to" is represented by ≥. Putting it all together, our complete mathematical problem transforms from a long sentence into a concise and powerful inequality: 5(x + 27) ≥ 6(x + 26). See? Not so scary when you take it one step at a time! This clear translation is the foundation for our next step: actually solving for 'x' and finding the solution set. It's a critical skill in mathematics and a prerequisite for any further steps in solving complex problems involving numerical relationships.

The Step-by-Step Solution: Unraveling the Inequality

Now that we've expertly translated our word problem into the algebraic inequality 5(x + 27) ≥ 6(x + 26), it's time to roll up our sleeves and solve for 'x'. This process involves using basic algebra rules, much like solving an equation, with one crucial distinction we'll get to soon. Our goal is to isolate 'x' on one side of the inequality symbol. Let's walk through it step-by-step to find our solution set.

Step 1: Distribute the numbers outside the parentheses. The first thing we need to do is apply the distributive property. This means multiplying the number outside each set of parentheses by each term inside the parentheses:

  • For the left side: 5 * (x + 27) becomes (5 * x) + (5 * 27), which simplifies to 5x + 135.
  • For the right side: 6 * (x + 26) becomes (6 * x) + (6 * 26), which simplifies to 6x + 156.

So, our inequality now looks like this: 5x + 135 ≥ 6x + 156.

Step 2: Gather like terms by moving variables to one side and constants to the other. We want all the 'x' terms on one side and all the constant numbers on the other. It's often a good idea to move the 'x' term that results in a positive coefficient for 'x' if possible, but it's not strictly necessary. Let's try subtracting 5x from both sides to keep our 'x' term positive on the right:

5x + 135 - 5x ≥ 6x + 156 - 5x

This simplifies to: 135 ≥ x + 156.

Now, let's move the constant term 156 from the right side to the left side by subtracting 156 from both sides:

135 - 156 ≥ x + 156 - 156

This simplifies to: -21 ≥ x.

Step 3: Interpret the final inequality. Our final result is -21 ≥ x. This statement means "-21 is greater than or equal to x." In other words, 'x' must be a number that is less than or equal to -21. If you prefer to have 'x' on the left side, you can flip the entire inequality, but remember that when you reverse the order of the terms, you must also reverse the direction of the inequality sign. So, -21 ≥ x is exactly the same as x ≤ -21. This is a critical point in solving inequalities because it defines the entire range of values that satisfy the original condition. We've successfully isolated our variable 'x', and now we can clearly see the range of values that make the original statement true. This result, x ≤ -21, directly leads us to defining the solution set for this mathematical problem.

Understanding the Solution Set: What Does x ≤ -21 Mean?

So, we've done the algebra and arrived at our elegant solution: x ≤ -21. But what does this really mean in the grand scheme of our original mathematical problem? The solution set for an inequality is the collection of all possible values for 'x' that make the inequality true. In our case, x ≤ -21 means that 'x' can be -21 itself, or any number that is smaller than -21. Think of it like this: if 'x' were -22, would the original statement hold true? Yes! If 'x' were -500, would it still hold true? Absolutely! But if 'x' were -20, or 0, or 100, the original statement would be false. This concept of a solution set is fundamental to mastering inequalities and helps us visualize the range of answers.

To represent this range of numbers concisely, mathematicians often use interval notation. For x ≤ -21, the interval notation is (-∞, -21]. Let's break down why: The ( symbol next to -∞ (negative infinity) means that there is no lower bound, and infinity is never included as a specific number. The ] symbol next to -21 means that -21 is included in the solution set. If -21 were not included (i.e., if it was strictly x < -21), we would use a parenthesis, like (-∞, -21). This specific notation helps us quickly communicate the exact boundaries and inclusivity of our solution.

You can also visualize this solution set on a number line. Imagine a straight line representing all numbers. You would place a closed circle (or a filled-in dot) at -21, because -21 is part of the solution. Then, you would draw an arrow extending to the left from that closed circle, indicating that all numbers smaller than -21 (down towards negative infinity) are also part of the solution. This visual representation can be incredibly helpful for understanding what x ≤ -21 truly signifies.

Comparing our result to the given options, we find that B. (-∞, -21] perfectly matches our derived solution set. This means that any number from negative infinity up to and including -21 will satisfy the original condition, making the statement "Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26" a true statement. Understanding this final step—translating your algebraic answer into a clear solution set and visualizing it—is the mark of true comprehension when solving inequalities.

Why Mastering Inequalities Matters Beyond the Classroom

While solving inequalities might seem like just another math exercise, the skills you develop are incredibly valuable and transcend the classroom walls. Mastering inequalities isn't just about passing a test; it's about developing a powerful toolset for navigating real-world challenges and making informed decisions. Think about situations where you need to consider a range of possibilities rather than a single fixed answer. That's where inequalities shine!

For instance, in personal finance and budgeting, you might set a rule: "My monthly expenses (E) must be less than or equal to my monthly income (I)." This translates to E ≤ I. Or perhaps you want to save at least $500 this year; your savings (S) must satisfy S ≥ 500. These are real-world applications of inequalities that help you manage your money effectively and achieve your financial goals. Without understanding these concepts, making sound financial plans would be much harder.

In engineering and design, inequalities are used to define constraints and safety margins. For example, the load a bridge can handle (L) must be less than or equal to its maximum capacity (C), so L ≤ C. A building's height (H) might need to be greater than or equal to a certain minimum for functionality, but less than or equal to another maximum for zoning, giving us a compound inequality like MinH ≤ H ≤ MaxH. These constraints ensure structures are safe, functional, and compliant with regulations. From civil engineering to designing microchips, mathematical problems involving limits and ranges are everywhere, and inequalities provide the language to express them.

Even in everyday decision-making, you implicitly use inequalities. "I need to leave for the airport no later than 2 PM" (time ≤ 2 PM). "I need at least 8 hours of sleep" (sleep ≥ 8 hours). These informal thought processes are foundational mathematical concepts that influence how we plan our day, manage our health, and interact with the world around us. By consciously understanding how to set up and solve these mathematical relations, you gain a clearer, more logical approach to problem-solving.

Ultimately, learning to translate complex statements into algebraic expressions and then systematically solve for variables within an inequality enhances your critical thinking skills. It teaches you to break down problems, analyze relationships, and think logically about potential outcomes. These are transferable skills that will serve you well, whether you're analyzing data, planning a project, or simply trying to make sense of the world. So, mastering solving inequalities isn't just a math lesson; it's a life skill!

Tips for Conquering Any Inequality Problem

Ready to tackle more inequalities with confidence? Here are some quick tips to help you on your journey:

  • Read Carefully: Always take your time to understand the problem statement. Identify keywords that indicate inequality symbols (e.g., "at least," "no more than," "greater than").
  • Translate Accurately: Convert word problems into algebraic expressions and inequalities step-by-step. Use parentheses correctly when a number multiplies an entire sum or difference.
  • Isolate the Variable: Use inverse operations to get the variable by itself, just like in equations. Remember to perform the same operation on both sides of the inequality.
  • Remember the Sign Flip Rule: This is crucial! If you multiply or divide both sides of an inequality by a negative number, you MUST flip the direction of the inequality sign. For example, if -2x > 6, dividing by -2 changes it to x < -3.
  • Check Your Answer: Pick a value within your solution set and one outside of it. Plug them back into the original inequality to see if they make the statement true or false as expected. This simple step can catch many errors.
  • Visualize: Use a number line to graph your solution set. It helps confirm your understanding of the interval and whether the endpoints are included or excluded.

Your Inequality Journey Continues!

Congratulations! You've successfully navigated the twists and turns of a complex inequality problem, transforming a wordy challenge into a clear, understandable solution. We started with the statement, "Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26," and through careful translation and step-by-step algebra, we discovered that the solution set is x ≤ -21, or (-∞, -21] in interval notation. This journey from problem statement to solution set showcases the power of algebra and the importance of precise thinking in mathematics.

Remember, mastering inequalities isn't just about memorizing rules; it's about understanding the logic behind them and appreciating their vast real-world applications. From personal finance to engineering, inequalities provide a framework for describing limits, ranges, and possibilities. By breaking down complex mathematical problems into smaller, manageable parts, you can build confidence and develop critical thinking skills that extend far beyond the math class.

Keep practicing, keep exploring, and don't be afraid to tackle new challenges. Each problem you solve is another step towards becoming a more confident and capable problem-solver. Your journey in mathematics is just beginning, and with each inequality you conquer, you're building a stronger foundation for future learning. Keep up the great work!

For more great resources and to continue your learning journey in algebra and inequalities, check out these trusted websites:

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