Solving 5x/(x-6) > 2 A Step-by-Step Guide To Rational Inequalities

Jul 10, 2025 by ADMIN 67 views

Solving the Rational Inequality: A Comprehensive Guide to 5x/(x-6) > 2

Rational inequalities, like the one we're tackling today, 5x/(x-6) > 2, might seem daunting at first glance. However, with a systematic approach, they become quite manageable. This article provides a comprehensive, step-by-step guide to solving this specific inequality and rational inequalities in general. We'll break down each step, explaining the reasoning behind it, so you'll not only get the answer but also understand the process. Mastering these techniques is crucial for success in algebra, calculus, and beyond.

Understanding Rational Inequalities

Before diving into the specifics, let's establish a solid foundation. A rational inequality is an inequality that involves a rational expression, which is simply a fraction where the numerator and/or the denominator are polynomials. Our inequality, 5x/(x-6) > 2, perfectly fits this definition. The key difference between solving rational inequalities and rational equations is that with inequalities, we need to consider intervals where the expression is either greater than, less than, greater than or equal to, or less than or equal to a specific value (in our case, 2). This requires a different strategy than simply finding specific solutions as we would with equations.

When dealing with inequalities, we can't simply multiply both sides by the denominator as we do with equations because the denominator, (x-6), could be positive or negative, and multiplying by a negative value would flip the inequality sign. This is a crucial point that necessitates a more careful approach. The most common method is to bring all terms to one side, combine them into a single rational expression, and then analyze the sign of the expression across different intervals.

Step 1: Rearranging the Inequality

The first crucial step in solving any rational inequality is to manipulate it so that one side is zero. This allows us to easily analyze the sign of the expression on the other side. In our case, we start with:

5x/(x-6) > 2

To get zero on the right side, we subtract 2 from both sides:

5x/(x-6) - 2 > 0

This rearrangement is paramount because it sets the stage for combining the terms into a single fraction, which is essential for the next step. By having zero on one side, we can focus solely on determining when the rational expression on the left is positive (greater than zero) or negative (less than zero).

Step 2: Combining into a Single Fraction

Now, we need to combine the terms on the left-hand side into a single fraction. To do this, we find a common denominator, which in this case is (x-6). We rewrite 2 as a fraction with this denominator:

5x/(x-6) - 2(x-6)/(x-6) > 0

Now we can combine the numerators:

(5x - 2(x-6))/(x-6) > 0

Next, we simplify the numerator by distributing the -2 and combining like terms:

(5x - 2x + 12)/(x-6) > 0

(3x + 12)/(x-6) > 0

This single fraction is much easier to analyze than the original expression. We've successfully transformed the inequality into a form where we can directly examine the sign of the expression based on the values of x.

Step 3: Finding Critical Values

Critical values are the linchpins for solving rational inequalities. They are the values of x that make either the numerator or the denominator of the rational expression equal to zero. These values are crucial because they divide the number line into intervals where the expression's sign remains constant. In essence, they are the points where the expression can potentially change from positive to negative or vice versa.

To find the critical values, we set both the numerator and the denominator equal to zero and solve for x:

Numerator: 3x + 12 = 0

Solving for x, we get:

3x = -12

x = -4
Denominator: x - 6 = 0

Solving for x, we get:

x = 6

So, our critical values are x = -4 and x = 6. These two points are the key to unlocking the solution of our inequality.

Step 4: Creating a Sign Chart

A sign chart is a powerful tool for visualizing how the sign of a rational expression changes across different intervals. It helps us systematically analyze the expression's behavior and identify the intervals where the inequality holds true. To create a sign chart, we follow these steps:

Draw a number line: Draw a horizontal line representing the real number line.
Mark the critical values: Locate the critical values we found in the previous step (-4 and 6) on the number line and mark them. These points divide the number line into intervals.
Choose test values: Select a test value within each interval. These values will help us determine the sign of the expression in that interval. For example, we could choose -5 (less than -4), 0 (between -4 and 6), and 7 (greater than 6).
Evaluate the expression: Substitute each test value into the simplified rational expression, (3x + 12)/(x-6), and determine the sign of the result. We only need the sign, not the exact value.
- For x = -5: (3(-5) + 12)/(-5-6) = (-15 + 12)/(-11) = (-3)/(-11) = + (positive)
- For x = 0: (3(0) + 12)/(0-6) = (12)/(-6) = - (negative)
- For x = 7: (3(7) + 12)/(7-6) = (21 + 12)/(1) = + (positive)
Fill in the sign chart: Write the sign of the expression in each interval on the number line. This visually represents where the expression is positive and negative.

Our sign chart will look like this:

Interval:     (-∞, -4)    (-4, 6)    (6, ∞)
Test Value:    -5         0         7
Sign:          +         -         +

Step 5: Determining the Solution

Now that we have our sign chart, we can easily determine the solution to the inequality (3x + 12)/(x-6) > 0. We are looking for the intervals where the expression is greater than zero, which means we want the intervals where the sign is positive. From our sign chart, we see that the expression is positive in the intervals (-∞, -4) and (6, ∞).

However, we need to be careful about the endpoints. The original inequality is a strict inequality ('>'), which means we don't include the values where the expression is equal to zero. The numerator is zero at x = -4, so we don't include -4 in our solution. The denominator is zero at x = 6, which means the expression is undefined at x = 6, so we definitely don't include 6 in our solution.

Therefore, the solution to the inequality 5x/(x-6) > 2 in interval notation is:

(-∞, -4) ∪ (6, ∞)

This means that any value of x less than -4 or greater than 6 will satisfy the original inequality.

Key Takeaways and Common Mistakes

Always rearrange the inequality so that one side is zero before combining terms.
Find the critical values by setting both the numerator and denominator equal to zero.
Use a sign chart to systematically analyze the sign of the expression in each interval.
Pay attention to the inequality sign ('>', '<', '≥', '≤') and whether to include the endpoints in the solution.
Remember that values that make the denominator zero are always excluded from the solution, as they make the expression undefined.
A common mistake is to multiply both sides of the inequality by the denominator without considering its sign. This can lead to an incorrect solution.

Conclusion

Solving rational inequalities requires a methodical approach. By following the steps outlined in this article – rearranging the inequality, combining into a single fraction, finding critical values, creating a sign chart, and carefully determining the solution – you can confidently tackle these types of problems. Remember to always double-check your work and pay close attention to the details, especially when dealing with inequalities. Practice makes perfect, so work through various examples to solidify your understanding. With consistent effort, you'll master the art of solving rational inequalities.

This comprehensive guide has provided you with the tools and knowledge to solve rational inequalities like 5x/(x-6) > 2. Now, go forth and conquer those inequalities!