Solving Inequality: Find F In 8 + 4(-f - 6) ≤ 4f - 6 - 2f

Have you ever encountered an inequality and felt a little lost on how to solve it? Don't worry, you're not alone! Inequalities might seem daunting at first, but with a systematic approach, they can be quite straightforward. In this article, we'll break down the process of solving the inequality $8 + 4(-f - 6) \leq 4f - 6 - 2f$ for the variable f, and we'll make sure to express the answer in its simplest form. So, grab a pen and paper, and let's dive in!

Understanding Inequalities

Before we jump into solving our specific problem, let's quickly recap what inequalities are. Unlike equations, which state that two expressions are equal, inequalities compare expressions using symbols like < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). Solving an inequality means finding the range of values for the variable that makes the inequality true. This range is often represented as an interval on the number line.

The core concept to grasp when dealing with inequalities is that operations performed on both sides must maintain the truth of the inequality. Most operations are similar to those used in equations, with one crucial exception: multiplying or dividing both sides by a negative number reverses the inequality sign. This is a critical rule to remember, as it directly impacts the solution.

Why is Solving Inequalities Important?

Solving inequalities isn't just an abstract mathematical exercise; it has practical applications in various fields. Inequalities are used in economics to model constraints and optimize solutions, in physics to describe the ranges of physical quantities, and in computer science to analyze algorithms and data structures. Understanding how to solve inequalities is a fundamental skill that empowers you to tackle real-world problems.

For example, consider a business trying to maximize profit. They might use inequalities to model constraints on resources like raw materials or production time. By solving these inequalities, they can determine the optimal production levels to achieve their profit goals. Similarly, in engineering, inequalities can be used to ensure that structures are strong enough to withstand certain loads or that electrical circuits operate within safe limits. This highlights the real-world importance of mastering the art of solving inequalities.

Step-by-Step Solution

Now, let's tackle our specific inequality: $8 + 4(-f - 6) \leq 4f - 6 - 2f$. We'll break it down step-by-step to make the process clear and easy to follow.

Step 1: Distribute

The first step is to distribute the 4 across the parentheses on the left side of the inequality. This means multiplying 4 by both -f and -6:

$8 + 4(-f - 6) \leq 4f - 6 - 2f$

$8 - 4f - 24 \leq 4f - 6 - 2f$

This simplifies the left side of the inequality by removing the parentheses and combining like terms, which sets the stage for further simplification and isolation of the variable.

Step 2: Combine Like Terms

Next, we'll combine the constant terms on the left side (8 and -24) and the 'f' terms on the right side (4f and -2f):

$8 - 4f - 24 \leq 4f - 6 - 2f$

$-4f - 16 \leq 2f - 6$

Combining like terms is a crucial step in simplifying any algebraic expression or inequality. It helps to consolidate terms, making the expression more manageable and easier to work with. This step reduces the complexity of the inequality, bringing us closer to isolating the variable 'f'.

Step 3: Isolate the Variable Term

Our goal is to get all the 'f' terms on one side of the inequality and the constant terms on the other. Let's add 4f to both sides:

$-4f - 16 \leq 2f - 6$

$-4f + 4f - 16 \leq 2f + 4f - 6$

$-16 \leq 6f - 6$

Isolating the variable term is a pivotal step in solving inequalities. By strategically adding or subtracting terms from both sides, we aim to gather all the terms containing the variable on one side, while moving the constants to the opposite side. This process simplifies the inequality, making it easier to determine the range of values for the variable that satisfy the condition.

Step 4: Isolate the Constant Term

Now, let's add 6 to both sides to isolate the constant term on the left side:

$-16 \leq 6f - 6$

$-16 + 6 \leq 6f - 6 + 6$

$-10 \leq 6f$

Similar to isolating the variable term, isolating the constant term is essential for simplifying the inequality and bringing it closer to its final solution. By adding or subtracting constants from both sides, we aim to group the constants on one side, allowing us to focus solely on the variable term and its coefficient.

Step 5: Solve for 'f'

To solve for 'f', we need to divide both sides by 6. Since we're dividing by a positive number, we don't need to reverse the inequality sign:

$-10 \leq 6f$

$\frac{-10}{6} \leq \frac{6f}{6}$

$-\frac{5}{3} \leq f$

Dividing both sides by the coefficient of the variable is the final step in isolating the variable and determining its range of values. In this case, we divide by 6, a positive number, which preserves the direction of the inequality. This step reveals the solution set for 'f', indicating the values that satisfy the original inequality.

Step 6: Express the Solution in Simplest Form

The solution $-\frac{5}{3} \leq f$ means that 'f' is greater than or equal to -5/3. We can also write this as:

$f \geq -\frac{5}{3}$

This is the simplest form of our solution. It clearly states the range of values for 'f' that satisfy the original inequality. Expressing the solution in its simplest form ensures clarity and conciseness, making it easier to interpret and apply in various contexts.

Checking the Solution

It's always a good idea to check your solution to make sure it's correct. We can do this by picking a value for 'f' that satisfies our inequality ($f \geq -\frac{5}{3}$) and plugging it back into the original inequality.

Let's choose $f = 0$, which is greater than $-\frac{5}{3}$:

$8 + 4(-0 - 6) \leq 4(0) - 6 - 2(0)$

$8 + 4(-6) \leq -6$

$8 - 24 \leq -6$

$-16 \leq -6$

This is true, so our solution is likely correct.

Checking the solution is a crucial step in the problem-solving process. By substituting a value within the solution range back into the original inequality, we can verify whether the inequality holds true. This process helps to catch any potential errors made during the solution process and ensures that the final answer is accurate and reliable.

Common Mistakes to Avoid

When solving inequalities, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution.

Forgetting to Reverse the Inequality Sign

The most common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This is a critical rule, and failing to apply it will lead to an incorrect solution.

Incorrect Distribution

Another common mistake is distributing incorrectly, especially when dealing with negative signs. Make sure to distribute the number correctly to each term inside the parentheses.

Combining Unlike Terms

It's important to only combine like terms. For example, you can't combine 'f' terms with constant terms. Mixing these up will lead to errors in your solution.

Arithmetic Errors

Simple arithmetic errors, such as adding or subtracting numbers incorrectly, can also lead to wrong answers. Take your time and double-check your calculations to minimize these errors.

Conclusion

Solving inequalities might seem challenging at first, but by following a systematic approach and being mindful of the rules, you can master this important skill. Remember to distribute, combine like terms, isolate the variable, and always check your solution. And, most importantly, don't forget to reverse the inequality sign when multiplying or dividing by a negative number!

By understanding the underlying principles and practicing regularly, you'll become more confident in your ability to solve inequalities and tackle more complex mathematical problems. So, keep practicing, and you'll be solving inequalities like a pro in no time!

For further learning and practice on inequalities, you can check out resources like Khan Academy's section on inequalities.