Solving 12/(x-2) < 6 A Step-by-Step Inequality Guide

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When tackling inequalities, especially those involving rational expressions, a methodical approach is crucial to ensure accurate solutions. This article delves into the process of solving the inequality 12x−2<6\frac{12}{x-2}<6, providing a step-by-step guide and explaining the underlying concepts. We will explore the importance of considering critical points and test intervals to arrive at the correct solution set. Understanding how to manipulate inequalities while preserving their validity is essential, and we will highlight common pitfalls to avoid. By the end of this guide, you will have a solid grasp of the techniques required to solve similar inequalities with confidence.

Understanding Inequalities forms the foundation of solving any mathematical problem of this type. Inequalities differ from equations in that they represent a range of possible values rather than a single solution. The inequality symbols (<, >, ≤, ≥) indicate whether a value is less than, greater than, less than or equal to, or greater than or equal to another value, respectively. When solving inequalities, the goal is to isolate the variable on one side of the inequality symbol, much like solving equations. However, there are specific rules to follow, particularly when dealing with multiplication or division by a negative number, which reverses the direction of the inequality. Additionally, when inequalities involve rational expressions, we must consider the values that make the denominator zero, as these values are excluded from the solution set. In the context of our problem, 12x−2<6\frac{12}{x-2}<6, we must carefully consider the value of x that makes the denominator (x - 2) equal to zero, as this will be a critical point in our solution. Ignoring these nuances can lead to incorrect solutions, emphasizing the importance of a systematic approach.

1. Initial Setup: Isolating the Rational Expression

Isolating the rational expression is a critical first step. Our initial inequality is 12x−2<6\frac{12}{x-2}<6. To begin, we aim to get all terms on one side, leaving zero on the other. This is achieved by subtracting 6 from both sides of the inequality:

12x−2−6<0\frac{12}{x-2} - 6 < 0

This step is essential because it sets the stage for combining the terms into a single fraction, which simplifies the analysis. By having zero on one side, we can easily determine the intervals where the expression is positive or negative. This approach is fundamental for solving inequalities, as it allows us to identify the regions on the number line that satisfy the given condition. The subtraction operation maintains the validity of the inequality, ensuring that the solutions we find are consistent with the original problem. Neglecting this initial setup can lead to difficulties in the subsequent steps, making it harder to find the correct solution set. Therefore, mastering this technique is vital for solving inequalities effectively.

2. Combining Terms: Finding a Common Denominator

Finding a common denominator is the next crucial step in simplifying the inequality. After subtracting 6 from both sides, we have 12x−2−6<0\frac{12}{x-2} - 6 < 0. To combine the terms, we need to express 6 as a fraction with the same denominator as the first term, which is (x - 2). This involves multiplying 6 by x−2x−2\frac{x-2}{x-2}, resulting in:

12x−2−6(x−2)x−2<0\frac{12}{x-2} - \frac{6(x-2)}{x-2} < 0

Now that both terms have the same denominator, we can combine them into a single fraction:

12−6(x−2)x−2<0\frac{12 - 6(x-2)}{x-2} < 0

This step is significant because it transforms the inequality into a simpler form where we can analyze the numerator and denominator more easily. By combining the terms, we reduce the complexity of the expression, making it easier to identify critical points and test intervals. The common denominator allows us to treat the inequality as a single rational expression, which is a standard form for solving such problems. Failing to find a common denominator would prevent us from simplifying the inequality further, hindering our ability to find the solution set.

3. Simplifying the Expression: Expanding and Combining Like Terms

Simplifying the expression is vital for revealing the structure of the inequality. We expand the numerator by distributing the -6 across the (x - 2) term:

12−6x+12x−2<0\frac{12 - 6x + 12}{x-2} < 0

Next, we combine like terms in the numerator:

24−6xx−2<0\frac{24 - 6x}{x-2} < 0

This simplification is essential because it presents the inequality in its most manageable form. By expanding and combining like terms, we reduce the complexity of the numerator, making it easier to identify the zeros of the expression. The simplified form allows us to analyze the behavior of the inequality more effectively, as we can now focus on the critical points where the numerator or denominator equals zero. This step is a fundamental algebraic technique that is crucial for solving a wide range of mathematical problems. Neglecting to simplify the expression would make it significantly harder to proceed with the solution, highlighting the importance of this step.

4. Identifying Critical Points: Zeros of Numerator and Denominator

Identifying critical points is a key step in solving inequalities, as these points divide the number line into intervals where the expression's sign remains constant. Critical points are the values of x that make either the numerator or the denominator equal to zero.

For the numerator, 24 - 6x = 0, we solve for x:

6x = 24

x = 4

For the denominator, x - 2 = 0, we solve for x:

x = 2

Thus, the critical points are x = 2 and x = 4. These points are crucial because they represent the values where the expression can change its sign. The critical points divide the number line into intervals, and within each interval, the expression will be either positive or negative. Therefore, identifying these points is essential for determining the solution set of the inequality. Failing to identify all critical points can lead to an incomplete or incorrect solution, emphasizing the importance of this step in the solving process.

5. Creating Test Intervals: Dividing the Number Line

Creating test intervals involves dividing the number line using the critical points we identified in the previous step. The critical points x = 2 and x = 4 divide the number line into three intervals:

  1. x < 2
  2. 2 < x < 4
  3. x > 4

These intervals are crucial because the sign of the expression 24−6xx−2\frac{24 - 6x}{x-2} will remain constant within each interval. This is because the expression can only change its sign at the critical points, where either the numerator or the denominator equals zero. By dividing the number line into these intervals, we can systematically test values within each interval to determine whether the inequality is satisfied. This method provides a structured approach to solving inequalities, ensuring that we consider all possible solutions. Neglecting to create these intervals would make it difficult to determine the solution set accurately, highlighting the importance of this step.

6. Testing Intervals: Determining the Sign of the Expression

Testing intervals is the process of selecting a test value within each interval and substituting it into the simplified inequality 24−6xx−2<0\frac{24 - 6x}{x-2} < 0 to determine the sign of the expression in that interval.

  1. For the interval x < 2:

    Let's choose x = 0 as a test value:

    24−6(0)0−2=24−2=−12<0\frac{24 - 6(0)}{0-2} = \frac{24}{-2} = -12 < 0 (True)

    Therefore, the inequality holds for x < 2.

  2. For the interval 2 < x < 4:

    Let's choose x = 3 as a test value:

    24−6(3)3−2=61=6>0\frac{24 - 6(3)}{3-2} = \frac{6}{1} = 6 > 0 (False)

    Therefore, the inequality does not hold for 2 < x < 4.

  3. For the interval x > 4:

    Let's choose x = 5 as a test value:

    24−6(5)5−2=−63=−2<0\frac{24 - 6(5)}{5-2} = \frac{-6}{3} = -2 < 0 (True)

    Therefore, the inequality holds for x > 4.

This step is essential because it allows us to identify the intervals where the inequality is satisfied. By testing values within each interval, we can determine whether the expression is positive or negative, and thus whether it meets the condition of being less than zero. The choice of test values is arbitrary, as the sign of the expression will remain constant within each interval. Failing to test these intervals would leave us unable to determine the solution set, emphasizing the importance of this step in the solving process.

7. Writing the Solution: Combining the Intervals

Writing the solution involves combining the intervals where the inequality holds true, based on our testing in the previous step. From our testing, we found that the inequality 24−6xx−2<0\frac{24 - 6x}{x-2} < 0 is satisfied for:

  • x < 2
  • x > 4

Therefore, the solution to the inequality is the union of these two intervals. In interval notation, this is expressed as:

x ∈ (-∞, 2) ∪ (4, ∞)

This solution set represents all the values of x that make the original inequality true. It is crucial to exclude x = 2 from the solution, as it makes the denominator of the original expression equal to zero, which is undefined. The solution set includes all values less than 2 and all values greater than 4. This step is the culmination of the solving process, where we synthesize the information gathered from previous steps to provide the final answer. Neglecting to write the solution accurately would leave the problem unfinished, highlighting the importance of this step.

8. Conclusion: Final Answer and Review

In conclusion, solving the inequality 12x−2<6\frac{12}{x-2}<6 involves a series of steps that include isolating the rational expression, finding a common denominator, simplifying the expression, identifying critical points, creating test intervals, testing those intervals, and finally, writing the solution. By following these steps meticulously, we arrive at the solution: x < 2 or x > 4, which in interval notation is x ∈ (-∞, 2) ∪ (4, ∞).

Reviewing the process is essential to ensure that no errors were made and that the solution is correct. This involves revisiting each step, checking the algebraic manipulations, and verifying that the critical points and test intervals were handled correctly. It is also a good practice to substitute values from the solution set back into the original inequality to confirm that they satisfy the condition. This final check provides confidence in the accuracy of the solution. Understanding and mastering these techniques not only helps in solving similar inequalities but also enhances problem-solving skills in mathematics in general.

Therefore, the correct answer is B. x < 2 or x > 4.

This step-by-step guide provides a comprehensive approach to solving inequalities involving rational expressions, equipping you with the skills and knowledge to tackle similar problems effectively.

Keywords

Solving inequalities, rational expressions, critical points, test intervals, solution set, algebraic manipulation, mathematical problem-solving, step-by-step guide, inequality symbols, numerator, denominator.