Solving Inequalities: 4 - (1/3)x > 11

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Solving Inequalities: $4-\frac{1}{3} x>11$

Hey guys, today we're diving deep into the nitty-gritty of solving inequalities, specifically tackling the problem: 4−13x>114-\frac{1}{3} x>11. This isn't just about crunching numbers; it's about understanding the logic behind why we do each step. Think of inequalities like a balancing act. Whatever you do to one side, you must do to the other to keep that balance. Our goal is to isolate 'x', kind of like getting your favorite video game all to yourself without any distractions. We want to know what values of 'x' make this statement true. It's a fundamental skill in mathematics that opens doors to more complex problem-solving, so let's break it down step-by-step. We'll go through each move, explaining the reasoning, and by the end, you'll be a pro at handling these types of problems. So, grab your favorite drink, settle in, and let's get this math party started!

Understanding the Inequality

Alright team, let's kick things off by really getting what this inequality, 4−13x>114-\frac{1}{3} x>11, is all about. The "greater than" symbol (>>) is key here. It means the expression on the left side (that's 4−13x4-\frac{1}{3} x) has to be larger than the number on the right side, which is 11. We're not looking for a single 'x' value like we would in an equation. Instead, we're looking for a range of 'x' values that satisfy this condition. Imagine you're trying to figure out how many hours you need to work to earn more than $11. If each hour you work gives you a certain amount of money (related to −13x-\frac{1}{3} x), and you start with $4, you want to find the minimum hours to exceed that $11 mark. The negative sign in front of the 13x\frac{1}{3} x is super important – it means as 'x' gets bigger, the value of that term actually gets smaller, which is a crucial detail we'll remember as we solve. Our primary mission, should we choose to accept it, is to get 'x' all by itself on one side of the inequality sign. This involves a series of inverse operations, kind of like a strategic game of chess where each move prepares for the next. We need to be mindful of the order of operations (PEMDAS/BODMAS) but working backward when isolating a variable. So, the first step in our grand strategy is to deal with that constant '4' sitting pretty next to our 'x' term. We want to move it over to the other side, leaving the 'x' term alone for a moment. This is where the concept of inverse operations shines. The inverse of adding 4 is subtracting 4, and as we've already established, whatever we do to one side, we must mirror on the other. This maintains the truthfulness of the inequality. So, get ready, because we're about to make that '4' disappear from the left side!

Step 1: Isolate the term with 'x'

Okay, team, let's get down to business and isolate the term that contains our beloved 'x'. Right now, in our inequality 4−13x>114-\frac{1}{3} x>11, the 'x' term (−13x-\frac{1}{3} x) is hanging out with a '+4'. To get this '-13x\frac{1}{3} x' by itself, we need to eliminate that '+4'. How do we do that? Easy peasy, lemon squeezy – we use the inverse operation! The opposite of adding 4 is subtracting 4. So, we're going to subtract 4 from both sides of the inequality. This is the golden rule, remember? Keep that balance!

Mathematically, it looks like this:

4−13x−4>11−44 - \frac{1}{3} x - 4 > 11 - 4

On the left side, the '+4' and the '-4' cancel each other out, leaving us with just our 'x' term: $ - \frac{1}{3} x $. On the right side, we perform the subtraction: 11−4=711 - 4 = 7.

So, our inequality now simplifies to:

$ - \frac{1}{3} x > 7 $

Boom! See? We've successfully isolated the term containing 'x'. It's still got that pesky negative sign and the fraction, but that's what our next step is all about. Give yourselves a pat on the back – you've conquered the first hurdle! This step is crucial because it separates the variable part from the constant part, setting the stage for the final push to find the value of 'x'. It's all about strategic simplification, making the problem more manageable with each move. We're getting closer and closer to that 'x' value, and the momentum is on our side. Keep that energy up, guys, because the next step is just as straightforward but requires a bit of extra attention due to that negative sign.

Step 2: Solve for 'x'

Alright, we've crushed Step 1 and now have $ - \frac{1}{3} x > 7 $. Our next mission, should we choose to accept it (and we totally do!), is to get 'x' completely alone. Right now, 'x' is being multiplied by −13-\frac{1}{3}. To undo multiplication, we use division. So, we need to divide both sides of the inequality by −13-\frac{1}{3}.

Here's the super important part, guys, pay close attention! When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. It's like the inequality gets confused by the negative and needs its direction changed. So, that '>' sign is going to flip around and become a '<' sign.

Let's do the math:

Divide both sides by −13-\frac{1}{3}:

$ \frac{-\frac{1}{3} x}{-\frac{1}{3}} < \frac{7}{-\frac{1}{3}} $

On the left side, dividing by itself cancels everything out, leaving us with just 'x'.

On the right side, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of −13-\frac{1}{3} is −3-3. So, we have 7×(−3)7 \times (-3).

7×(−3)=−217 \times (-3) = -21

So, our final inequality becomes:

x<−21x < -21

And there you have it! We've successfully solved for 'x'. This means that any value of 'x' that is less than -21 will make the original inequality 4−13x>114 - \frac{1}{3} x > 11 true. Pretty cool, right? This is the power of understanding how to manipulate inequalities. The flipping of the sign when multiplying or dividing by a negative is a crucial detail that can trip people up, but now you know the secret! You've done an amazing job following along, and you should feel confident tackling similar problems. Remember this principle: isolate the variable term, then isolate the variable itself, always keeping the inequality sign's rules in mind, especially the negative number rule.

Conclusion: What Does x<−21x < -21 Mean?

So, we've landed on the solution x<−21x < -21. What does this actually mean in plain English, guys? It means that for the original statement 4−13x>114 - \frac{1}{3} x > 11 to be true, 'x' cannot be just any number. It has to be a number that is strictly less than -21. Think about it: if you plug in a number bigger than -21, say -20, let's see what happens: 4−13(−20)=4+203=4+6.67=10.674 - \frac{1}{3}(-20) = 4 + \frac{20}{3} = 4 + 6.67 = 10.67. Is 10.67>1110.67 > 11? Nope, it's not. That's why numbers greater than or equal to -21 don't work.

Now, let's try a number that is less than -21, like -24: 4−13(−24)=4+243=4+8=124 - \frac{1}{3}(-24) = 4 + \frac{24}{3} = 4 + 8 = 12. Is 12>1112 > 11? You bet it is! This confirms our solution. The inequality x<−21x < -21 represents an infinite set of numbers, all the numbers stretching out to negative infinity on the number line, stopping just before -21. It’s like saying, "Any number smaller than -21 is a winner!" This understanding of solution sets is fundamental in mathematics and is used everywhere, from graphing functions to analyzing data. You've navigated the complexities of inequalities, remembered the critical rule about negative multipliers, and arrived at a clear, concise solution. Congratulations on mastering this problem! Keep practicing, and these concepts will become second nature. You guys are doing great!