Solve: 5(x+27) >= 6(x+26)

Hey guys! Today we're diving into a super interesting problem that might seem a bit wordy at first, but trust me, once we break it down, it's a piece of cake. We're tackling an inequality: "Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26." Our mission, should we choose to accept it, is to find the solution set for this mathematical puzzle. This means we need to figure out all the possible values of the 'number' that make this statement true. Inequalities like these are super common in algebra, and understanding how to solve them is a fundamental skill. They pop up everywhere, from figuring out the minimum profit a business needs to make to determining the range of values a certain measurement can take. So, buckle up, grab your favorite beverage, and let's get this solved!

Understanding the Inequality: Translating Words into Math

Alright, first things first, let's translate this beast of a sentence into a mathematical expression. When we talk about "a number," we usually represent it with a variable, let's use 'x' because it's a classic. The phrase "the sum of a number and 27" translates directly to (x + 27). Now, "five times the sum of a number and 27" becomes 5 * (x + 27). Easy peasy, right? Next up, we have "six times the sum of that number and 26." Following the same logic, this translates to 6 * (x + 26). Finally, the key phrase is "is greater than or equal to." In math-speak, this is represented by the symbol >=. So, putting it all together, our inequality looks like this: 5(x + 27) >= 6(x + 26). This is the core of our problem, and solving it will reveal the magical solution set. Remember, the goal is to isolate 'x' and see what values it can take.

Step-by-Step Solution: Cracking the Inequality

Now for the fun part – solving the inequality! We've got 5(x + 27) >= 6(x + 26). The first step in simplifying this is to distribute the numbers outside the parentheses to the terms inside. So, on the left side, we multiply 5 by x and 5 by 27, giving us 5x + 135. On the right side, we multiply 6 by x and 6 by 26, which results in 6x + 156. Our inequality now looks like this: 5x + 135 >= 6x + 156. The next move is to get all the 'x' terms on one side and all the constant numbers on the other. It doesn't strictly matter which side you choose for the 'x's, but often it's easier to keep the 'x' coefficient positive. Let's subtract 5x from both sides. This gives us 135 >= x + 156. Now, we need to get 'x' by itself. So, we subtract 156 from both sides: 135 - 156 >= x. Performing the subtraction, we get -21 >= x. This means that 'x' must be less than or equal to -21. So, any number that is -21 or smaller will satisfy our original inequality. This is super cool because it tells us a whole range of numbers works, not just one specific number.

Interpreting the Solution Set: What Does It Mean?

We've found that x <= -21. But what does this actually mean in terms of a solution set? A solution set is simply the collection of all possible values for the variable (in our case, 'x') that make the inequality true. So, the inequality x <= -21 tells us that any number that is less than or equal to -21 is a valid solution. This includes numbers like -21, -21.5, -22, -100, and so on, all the way down to negative infinity. When we represent this on a number line, we would have a closed circle at -21 (because -21 itself is included in the solution) and an arrow pointing to the left, indicating all numbers less than -21. The notation for this in interval form is (-∞, -21]. The parenthesis on the left signifies that negative infinity is not a number we can actually reach, while the square bracket on the right indicates that -21 is included in our set. This is how mathematicians concisely communicate all the numbers that work in our original problem. It’s a powerful way to express an infinite number of solutions!

Checking Our Work: Ensuring Accuracy

It's always a smart move in math, especially when dealing with inequalities, to check our answer. We found that the solution set is x <= -21. Let's pick a value within this set and see if it works. How about x = -22? Plugging this back into our original inequality 5(x + 27) >= 6(x + 26):

Left side: 5(-22 + 27) = 5(5) = 25

Right side: 6(-22 + 26) = 6(4) = 24

Is 25 >= 24? Yes, it is! So, x = -22 works. Now, let's test a value outside our solution set. How about x = -20 (which is greater than -21)?

Left side: 5(-20 + 27) = 5(7) = 35

Right side: 6(-20 + 26) = 6(6) = 36

Is 35 >= 36? No, it is not! This confirms that values greater than -21 do not satisfy the inequality. Lastly, let's check the boundary case, x = -21 itself.

Left side: 5(-21 + 27) = 5(6) = 30

Right side: 6(-21 + 26) = 6(5) = 30

Is 30 >= 30? Yes, it is! This confirms that -21 is indeed part of our solution set. Our checks all line up, giving us confidence in our answer.

Final Answer: The Solution Set Revealed

So, after all that number crunching and algebraic maneuvering, we've arrived at the solution. The inequality 5(x + 27) >= 6(x + 26) simplifies to x <= -21. This means that any number less than or equal to -21 will make the original statement true. When we express this as a solution set, we're looking for the option that represents all numbers from negative infinity up to and including -21. Looking at the options provided:

A. $(-\infty, -21)$ - This excludes -21. B. $(-\infty, -21]$ - This includes -21 and all numbers less than it. C. $[-21, +\infty)$ - This includes -21 and all numbers greater than it. D. $(21, +\infty)$ - This represents numbers greater than 21.

Based on our calculations, the correct solution set is (-\infty, -21]. So, the answer is B. It's awesome how we can take a word problem, transform it into math, solve it, and then interpret the results. Keep practicing these, guys, and you'll be inequality wizards in no time!