Solving The Inequality 8000 - 200x ≥ 5000 + 300x A Step-by-Step Guide

In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries for variables. Understanding how to solve inequalities is fundamental for various applications, from real-world problem-solving to advanced mathematical concepts. In this comprehensive guide, we will delve into the step-by-step process of solving the inequality $8,000 - 200x \geq 5,000 + 300x$. This inequality represents a linear relationship where we aim to find the values of 'x' that satisfy the given condition. By mastering the techniques involved in solving this inequality, you will gain valuable skills applicable to a wide range of mathematical scenarios.

Understanding Inequalities

Before we dive into the specifics of solving the given inequality, let's establish a firm understanding of what inequalities represent. Inequalities are mathematical expressions that compare two values, indicating that one value is either greater than, less than, greater than or equal to, or less than or equal to the other value. Unlike equations, which seek to find specific values that make the two sides equal, inequalities define a range of values that satisfy the relationship. The symbols used in inequalities are:

• '>' (greater than)
• '<' (less than)
• '$\geq$' (greater than or equal to)
• '$\leq$' (less than or equal to)

When solving inequalities, our goal is to isolate the variable on one side of the inequality symbol, just as we do with equations. However, there's a crucial difference: when we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality symbol. This is because multiplying or dividing by a negative number changes the sign of the values, and thus the relationship between them. This concept is fundamental to accurately solving inequalities.

Step-by-Step Solution

Now, let's embark on the journey of solving the inequality $8,000 - 200x \geq 5,000 + 300x$ step by step. Our objective is to isolate the variable 'x' on one side of the inequality symbol. To achieve this, we will employ a series of algebraic manipulations, ensuring we maintain the integrity of the inequality throughout the process.

Step 1: Combine Like Terms

The initial step involves simplifying the inequality by combining like terms. This means grouping the constant terms (numbers without variables) together and the terms with 'x' together. To do this, we can subtract 5,000 from both sides of the inequality:

$8,000 - 200x - 5,000 \geq 5,000 + 300x - 5,000$

This simplifies to:

$3,000 - 200x \geq 300x$

Next, we want to move the term with 'x' from the left side to the right side. We can achieve this by adding 200x to both sides:

$3,000 - 200x + 200x \geq 300x + 200x$

This simplifies to:

$3,000 \geq 500x$

Step 2: Isolate the Variable

Now that we have all the 'x' terms on one side and the constant terms on the other, we need to isolate 'x'. To do this, we will divide both sides of the inequality by the coefficient of 'x', which is 500:

$\frac{3,000}{500} \geq \frac{500x}{500}$

This simplifies to:

$6 \geq x$

Step 3: Express the Solution

The solution to the inequality is $6 \geq x$, which can also be written as $x \leq 6$. This means that any value of 'x' that is less than or equal to 6 will satisfy the original inequality. We can represent this solution graphically on a number line, where the region to the left of 6 (including 6) is shaded to indicate the solution set.

Verification

To ensure the accuracy of our solution, it's always a good practice to verify it. We can do this by selecting a value within the solution set (i.e., a value less than or equal to 6) and substituting it back into the original inequality. If the inequality holds true, our solution is likely correct. For example, let's choose x = 0:

$8,000 - 200(0) \geq 5,000 + 300(0)$

$8,000 \geq 5,000$

This is true, which supports our solution.

Let's also test a value outside the solution set, say x = 7:

$8,000 - 200(7) \geq 5,000 + 300(7)$

$8,000 - 1,400 \geq 5,000 + 2,100$

$6,600 \geq 7,100$

This is false, confirming that our solution set is correct.

Graphical Representation

Visualizing the solution on a number line provides a clear understanding of the range of values that satisfy the inequality. The number line extends infinitely in both directions, representing all real numbers. To represent the solution $x \leq 6$, we draw a closed circle (or a filled-in dot) at 6 on the number line, indicating that 6 is included in the solution set. Then, we shade the portion of the number line to the left of 6, representing all values less than 6. This shaded region represents the solution set for the inequality.

If the inequality were $x < 6$, we would use an open circle (or a hollow dot) at 6 to indicate that 6 is not included in the solution set. The shading to the left would still represent all values less than 6.

Real-World Applications

Inequalities are not just abstract mathematical concepts; they have numerous applications in real-world scenarios. Consider a business setting where a company wants to determine the number of units they need to sell to achieve a certain profit target. Inequalities can be used to model the relationship between sales, costs, and profits, allowing the company to determine the minimum number of units that must be sold to meet their financial goals. For example, if the profit is represented by P, the cost by C, the revenue per unit by R, and the number of units sold by x, an inequality could be set up as:

$P \geq Rx - C$

Solving this inequality for x would give the minimum number of units that need to be sold to achieve the desired profit P.

In other fields, such as engineering, inequalities are used to define safety margins and tolerances. For instance, when designing a bridge, engineers use inequalities to ensure that the structure can withstand a certain load with a specified safety factor. This ensures that the bridge can safely handle the expected traffic and environmental conditions.

Inequalities are also used in optimization problems, where the goal is to find the best possible solution within a set of constraints. These constraints are often expressed as inequalities, defining the feasible region within which the optimal solution must lie. This is commonly used in logistics, resource allocation, and financial planning.

Common Mistakes and How to Avoid Them

Solving inequalities involves careful attention to detail, and certain mistakes can lead to incorrect solutions. One of the most common errors is forgetting to reverse the inequality symbol when multiplying or dividing both sides by a negative number. This can completely change the solution set and lead to wrong conclusions. To avoid this, always double-check the sign of the number you are multiplying or dividing by and remember to flip the inequality symbol if it's negative.

Another common mistake is incorrectly combining like terms. Ensure that you are only adding or subtracting terms that have the same variable and exponent. For instance, you cannot combine a term with 'x' with a constant term. Organize your terms carefully to avoid these errors.

Careless arithmetic mistakes can also lead to incorrect solutions. It's essential to perform each step meticulously and double-check your calculations. If possible, use a calculator to verify your results, especially when dealing with larger numbers or complex expressions.

Finally, failing to verify your solution can lead to overlooking errors. Always substitute a value from your solution set (and a value outside the solution set) back into the original inequality to ensure it holds true. This simple step can help you catch mistakes and gain confidence in your answer.

Conclusion

In conclusion, solving the inequality $8,000 - 200x \geq 5,000 + 300x$ involves a series of algebraic steps aimed at isolating the variable 'x'. By combining like terms, isolating the variable, and expressing the solution in the correct format, we can accurately determine the range of values that satisfy the inequality. Understanding inequalities is a fundamental skill in mathematics, with applications spanning various fields. Remember to pay close attention to the rules of inequalities, especially the reversal of the inequality symbol when multiplying or dividing by a negative number. By mastering these techniques, you will be well-equipped to tackle a wide range of mathematical problems involving inequalities.

Solving inequalities, like the one we addressed, is a core skill in mathematics. The ability to manipulate and solve inequalities opens doors to advanced concepts and real-world applications. Remember to practice regularly and apply these techniques to different scenarios to strengthen your understanding. The solution $x \leq 6$ represents a set of values, and visualizing this on a number line can provide a deeper insight into the solution set. Keep exploring the world of mathematics, and you'll find that inequalities are powerful tools for problem-solving and decision-making. The key takeaways from this guide are the steps involved in solving inequalities, the importance of reversing the inequality symbol, and the practical applications of inequalities in various fields. With consistent practice and a solid understanding of the principles, you can confidently tackle any inequality that comes your way.