Solving The Inequality (3x+8)/(x-4) >= 0 A Step-by-Step Guide

When faced with inequalities, especially those involving rational expressions, a systematic approach is essential to arrive at the correct solution. In this comprehensive guide, we will delve into the step-by-step process of solving the inequality $\frac{3x+8}{x-4} \geq 0$. This type of problem is a cornerstone of algebra and calculus, and mastering it will enhance your problem-solving skills in various mathematical contexts. Our focus will be on providing a clear, detailed explanation that is accessible to students and anyone interested in brushing up on their math skills. We will cover the critical concepts, potential pitfalls, and strategies for ensuring accuracy. Let's embark on this mathematical journey together and unlock the secrets of inequality solutions.

Step 1: Identify Critical Points

The journey to solving this inequality begins with pinpointing the critical values of x that make either the numerator or the denominator equal to zero. These critical points are the cornerstones of our solution because they demarcate intervals on the number line where the expression's sign may change. Let's dissect the given inequality, $\frac{3x+8}{x-4} \geq 0$, and unearth these crucial values. First, we'll tackle the numerator, setting $3x + 8$ equal to zero. This gives us a linear equation, $3x + 8 = 0$. Solving for x involves a simple algebraic maneuver: subtracting 8 from both sides yields $3x = -8$, and then dividing by 3 gives us $x = -\frac{8}{3}$. This is our first critical point, a value where the expression could potentially switch signs. Next, we turn our attention to the denominator, $x - 4$. Setting this equal to zero, we get $x - 4 = 0$. Adding 4 to both sides, we find $x = 4$. This is our second critical point, a value that is particularly important because it makes the denominator zero, thus making the entire expression undefined. These two critical points, $-\frac{8}{3}$ and 4, are the key landmarks we'll use to map out our solution. They divide the number line into distinct intervals, each of which we'll analyze to determine where the inequality holds true. Understanding the significance of these points is paramount to successfully navigating the intricacies of inequality problems.

Step 2: Create a Sign Chart

Creating a sign chart is a pivotal step in solving inequalities, especially those involving rational expressions. This visual tool allows us to systematically analyze the behavior of the expression $\frac{3x+8}{x-4}$ across different intervals on the number line. Our critical points, $-\frac{8}{3}$ and 4, serve as the dividers, splitting the number line into three distinct intervals: $(-\infty, -\frac{8}{3})$, $(-\frac{8}{3}, 4)$, and $(4, \infty)$. Within each interval, the expression will maintain a consistent sign – either positive or negative – making our task of evaluation much more manageable. To construct the sign chart, we draw a number line and mark our critical points. Then, we select a test value within each interval. For the interval $(-\infty, -\frac{8}{3})$, we might choose $x = -3$. For $(-\frac{8}{3}, 4)$, a convenient choice is $x = 0$. And for $(4, \infty)$, we could pick $x = 5$. We then substitute each test value into the expression $\frac{3x+8}{x-4}$ and record the resulting sign. For $x = -3$, we get $\frac{3(-3)+8}{-3-4} = \frac{-1}{-7} = \frac{1}{7}$, which is positive. For $x = 0$, we have $\frac{3(0)+8}{0-4} = \frac{8}{-4} = -2$, which is negative. And for $x = 5$, we get $\frac{3(5)+8}{5-4} = \frac{23}{1} = 23$, which is positive. By recording these signs in our chart, we gain a clear picture of where the expression is positive, negative, or zero. This chart is not just a tool; it's a roadmap that guides us to the solution of the inequality. It transforms a potentially complex problem into a series of straightforward sign evaluations, making the solution process both transparent and reliable. The sign chart is an indispensable asset in our mathematical toolkit for tackling inequalities.

Step 3: Determine the Solution

After meticulously constructing our sign chart, we arrive at the crucial step of determining the solution to the inequality $\frac{3x+8}{x-4} \geq 0$. The sign chart acts as our compass, guiding us to the intervals where the expression is either positive or zero, which are the regions that satisfy our inequality. Recall that our critical points, $-\frac{8}{3}$ and 4, have divided the number line into three intervals: $(-\infty, -\frac{8}{3})$, $(-\frac{8}{3}, 4)$, and $(4, \infty)$. Our sign chart has revealed the sign of the expression within each of these intervals. We are looking for intervals where the expression is greater than or equal to zero, meaning we are interested in regions where the sign is either positive or zero. From our sign chart, we identify that the expression is positive in the intervals $(-\infty, -\frac{8}{3})$ and $(4, \infty)$. This means that any x value within these intervals will satisfy the inequality. However, we must also consider the points where the expression equals zero. The numerator, $3x + 8$, becomes zero when $x = -\frac{8}{3}$. Since our inequality includes "or equal to," this point is part of our solution. The denominator, $x - 4$, becomes zero when $x = 4$. However, because a zero denominator makes the expression undefined, $x = 4$ cannot be included in our solution, even though it makes the expression neither positive nor negative. Therefore, our solution includes the interval $(-\infty, -\frac{8}{3}]$ (note the square bracket indicating inclusion of $-\frac{8}{3}$) and the interval $(4, \infty)$ (note the parenthesis indicating exclusion of 4). Combining these, we express the solution in interval notation as $x \in (-\infty, -\frac{8}{3}] \cup (4, \infty)$. This comprehensive solution encapsulates all x values that satisfy the inequality, carefully accounting for both positive regions and critical points. It's a testament to the power of the sign chart method in providing a clear and accurate pathway to solving inequalities.

Conclusion: The Solution and Its Implications

In conclusion, the solution to the inequality $\frac{3x+8}{x-4} \geq 0$ is $x \leq -\frac{8}{3}$ or $x > 4$. This solution, meticulously derived through our step-by-step process, underscores the importance of a systematic approach when dealing with rational inequalities. We began by identifying the critical points, the values of x that nullify the numerator or denominator, as these points are the anchors around which the expression's sign may fluctuate. Then, we harnessed the power of the sign chart, a visual aid that allowed us to map the sign of the expression across the intervals defined by our critical points. This chart transformed a potentially daunting problem into a series of straightforward sign evaluations, revealing where the expression was positive, negative, or zero. Finally, we synthesized the information from the sign chart, carefully considering whether to include or exclude critical points based on the inequality's condition and the expression's definition. Our solution, expressed in interval notation as $x \in (-\infty, -\frac{8}{3}] \cup (4, \infty)$, encapsulates all x values that satisfy the inequality, providing a complete and accurate answer. This journey through the solution process highlights not just the mechanics of solving inequalities, but also the underlying logic and reasoning that empower us to tackle a wide range of mathematical challenges. Understanding these principles is key to success in algebra, calculus, and beyond.

Final Answer: The final answer is $\boxed{x \leq-\frac{8}{3} \text{ or } x>4}$