Solving Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Ready to dive into the world of inequalities? Today, we're going to tackle a problem that involves writing and solving an inequality. We'll be working with fractions, so buckle up, it's going to be a fun ride! This guide will break down the process step-by-step, making it super easy to understand. We will use the original input: "$-\frac{1}{3}$ is greater than or equal to the product of $-\frac{4}{5}$ and a number" to fully grasp the concepts.

Understanding the Problem: Deciphering the Inequality

Let's break down the problem statement: "-1/3 is greater than or equal to the product of -4/5 and a number." This sentence is packed with mathematical information, so let's translate it into symbols. First, we have the phrase "-rac{1}{3} is greater than or equal to", which we can represent using the symbols ≥. This symbol means that the value on the left side is either larger than or equal to the value on the right side. Next, we have "the product of -4/5 and a number." In math, "product" means multiplication. So, we're multiplying -rac{4}{5} by some unknown number. Let's represent this unknown number with the variable 'x'. Therefore, the phrase "the product of -4/5 and a number" can be written as -rac{4}{5}x or simply -rac{4x}{5}.

So, putting it all together, the inequality we need to write is -rac{1}{3} ≥ -rac{4}{5}x. This inequality states that -1/3 is greater than or equal to the product of -4/5 and the unknown number x. Understanding the different parts of the original statement is key to solving this type of inequality. This initial setup is crucial for correctly solving the inequality. The hardest part for most people is setting it up. Making sure you understand each word and what it means is key. It's like a secret code, and we've just cracked it! So now that we've successfully translated the problem into mathematical terms, we're ready to move on to the next step: solving the inequality. This step-by-step approach ensures that you grasp the concepts and build confidence in your ability to solve similar problems. Remember, practice makes perfect, so don't hesitate to work through more examples to solidify your understanding. The more you practice, the easier it will become to translate word problems into mathematical expressions. Stay focused, and take your time, and you'll become an inequality master in no time!

Solving the Inequality: Finding the Value of 'x'

Now, let's solve the inequality -rac1}{3} ≥ -rac{4}{5}x for x. Our goal is to isolate 'x' on one side of the inequality. Here's how we'll do it step-by-step. Firstly, to isolate 'x', we need to get rid of the -rac{4}{5} that's multiplying it. We can do this by multiplying both sides of the inequality by the reciprocal of -rac{4}{5}, which is -rac{5}{4}. **Important Note** When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. So, our inequality -rac{1{3} ≥ -rac{4}{5}x becomes:

(-rac{1}{3}) * (-rac{5}{4}) ≤ (-rac{4}{5}x) * (-rac{5}{4})

Multiplying the fractions on the left side, we get:

rac{5}{12} ≤ x

So, x ≤ rac{5}{12}. This is our solution! It means that any number less than or equal to 5/12 will satisfy the original inequality. In this step, we've carefully isolated x, following the rules of inequalities. Remember to flip the inequality sign whenever you multiply or divide by a negative number. This is a common mistake, so always double-check. The solution to the inequality gives us a range of values that satisfy the original statement. Think of it like this: Any number within this range will make the original statement true. Mastering this process is key to your success in algebra and other areas of mathematics. The ability to manipulate and solve inequalities is a fundamental skill that opens up a world of possibilities in problem-solving. Keep practicing, stay focused, and you'll get it!

Checking the Solution: Verification and Interpretation

To ensure our solution is correct, let's choose a value for 'x' that satisfies x ≤ rac{5}{12} and substitute it back into the original inequality. Let's choose x = 0 (since 0 is less than rac{5}{12}). Substituting x = 0 into -rac{1}{3} ≥ -rac{4}{5}x, we get:

-rac{1}{3} ≥ -rac{4}{5}(0)

-rac{1}{3} ≥ 0

Well, that's not exactly true, -1/3 is not greater than or equal to 0, which tells us that 0 cannot be an answer. Let's try to verify if 1/12 is an answer.

-rac{1}{3} ≥ -rac{4}{5}(rac{1}{12})

-rac{1}{3} ≥ -rac{4}{60}

-rac{1}{3} ≥ -rac{1}{15}

-rac{1}{3} is not greater than or equal to -rac{1}{15}, so this is not a valid solution.

So it means we made a mistake somewhere, let's recap our steps

We start with: -rac{1}{3} ≥ -rac{4}{5}x

Multiply by -5/4 to both sides:

(-rac{1}{3}) * (-rac{5}{4}) ≤ x

rac{5}{12} ≤ x

So x ≥ rac{5}{12} This means that any number greater than or equal to 5/12 will satisfy the original inequality. Now let's try again. Let's choose x = 1 (since 1 is greater than 5/12). Substituting x = 1 into -rac{1}{3} ≥ -rac{4}{5}x, we get:

-rac{1}{3} ≥ -rac{4}{5}(1)

-rac{1}{3} ≥ -rac{4}{5}

-rac{1}{3} = -0.33, and -rac{4}{5} = -0.8. So, -0.33 ≥ -0.8 which is a correct statement.

This confirms that our solution x ≥ rac{5}{12} is correct. Checking your solution is an essential step in problem-solving. It helps to catch any mistakes you might have made along the way. In this case, by plugging in a value for 'x' and evaluating the original inequality, we were able to confirm that our solution is valid. This process not only verifies the accuracy of your solution but also helps to deepen your understanding of the concept. Remember, always take the time to check your work; it's a valuable habit that will serve you well in all areas of mathematics. By checking our solution, we've gained confidence in our ability to solve the inequality and understand its implications. This is the power of verification; it transforms a solution from a mere answer to a solid understanding.

Conclusion: Mastering Inequalities

Congratulations, you've successfully written and solved an inequality! You've learned how to translate a word problem into a mathematical expression, solve for the unknown variable, and check your solution to ensure its accuracy. Remember, the key to mastering inequalities is practice. Work through various examples, pay attention to the rules (especially flipping the inequality sign), and don't be afraid to ask for help if you get stuck. You've got this! Keep practicing and challenging yourself with more complex problems. With each problem you solve, your understanding of inequalities will grow, and you'll become more confident in your ability to tackle any mathematical challenge. So, keep up the great work! The skills you've developed in solving inequalities are fundamental to many areas of mathematics and beyond. As you continue your mathematical journey, you'll find that these skills will serve you well, opening doors to more advanced concepts and problem-solving techniques. You've got the tools; now go out there and conquer those inequalities!