Which Inequality Is Equivalent To X-6/x 5 X 7/x 3

Decoding the Inequality: x - 6/x ≤ x + 7/x - 3

This article delves into the intricacies of solving the inequality x - 6/x ≤ x + 7/x - 3. We'll break down the process step-by-step, explaining the underlying mathematical principles and offering strategies for similar problems. Understanding inequalities is crucial for various mathematical applications, from calculus to linear programming. This detailed explanation aims to equip you with the tools to confidently tackle such problems.

Understanding the Problem

The inequality presented, x - 6/x ≤ x + 7/x - 3, involves rational expressions (fractions containing variables). Solving this requires a methodical approach to avoid errors. The core challenge lies in manipulating the fractions and isolating 'x' while maintaining the integrity of the inequality sign. We need to find the range of values for 'x' that satisfy the given condition.

Step 1: Simplify the Inequality

Our first step involves simplifying the inequality by combining like terms and eliminating fractions. Notice that 'x' appears on both sides of the inequality. Let's begin by subtracting 'x' from both sides:

x - 6/x - x ≤ x + 7/x - 3 - x

This simplifies to:

-6/x ≤ 7/x - 3

Next, let's isolate the terms with 'x' in the denominator. Add 6/x to both sides:

0 ≤ 7/x + 6/x - 3

Now, combine the fractions on the right-hand side:

0 ≤ (7 + 6)/x - 3

This further simplifies to:

0 ≤ 13/x - 3

Step 2: Isolate the Variable Term

To solve for 'x', let's isolate the term containing 'x'. Add 3 to both sides:

3 ≤ 13/x

Now, to get 'x' out of the denominator, we can reciprocate both sides. However, it's crucial to remember that when reciprocating both sides of an inequality, the inequality sign reverses if the values being reciprocated are negative. We must therefore consider two cases:

Case 1: x > 0

If x is positive, reciprocating the inequality maintains the direction of the inequality sign:

x/13 ≤ 1/3

Multiplying both sides by 13, we get:

x ≤ 13/3

Since we assumed x > 0 in this case, the solution for this case is:

0 < x ≤ 13/3

Case 2: x < 0

If x is negative, reciprocating reverses the inequality sign:

x/13 ≥ 1/3

Multiplying both sides by 13, we get:

x ≥ 13/3

This seems contradictory because we assumed x < 0, and 13/3 is positive. Therefore, there are no solutions in this case that satisfy both conditions.

Step 3: Combining the Solutions

Combining the solutions from both cases, we find that the solution to the inequality x - 6/x ≤ x + 7/x - 3 is:

0 < x ≤ 13/3

This means that the inequality holds true for all values of 'x' strictly greater than 0 and less than or equal to 13/3.

Graphical Representation

To visualize the solution, consider plotting the inequality on a number line. You would see a shaded region representing the interval (0, 13/3], indicating that 'x' can take on any value within this range, including 13/3 but excluding 0.

Handling Undefined Cases

It's crucial to acknowledge that the original inequality contains fractions with 'x' in the denominator. This means that 'x' cannot equal zero, as division by zero is undefined. Therefore, x = 0 is excluded from the solution set.

Advanced Considerations: Analyzing the Behavior of the Inequality

We can gain a deeper understanding of the inequality by analyzing the behavior of the expressions involved. Consider plotting the functions y = x - 6/x and y = x + 7/x - 3 on a graph. Observing the points where these functions intersect and comparing their values can provide further insight into the solution set. Such analysis can enhance your intuitive understanding of the inequality's behavior.

Similar Problem Solving Strategies

The approach outlined above can be applied to solve a broader range of inequalities involving rational expressions. The key steps remain consistent:

1. Simplify: Combine like terms and manipulate the inequality to isolate the variable.
2. Isolate the Variable: Use algebraic operations to isolate the term containing the variable.
3. Consider Cases: Pay careful attention to the signs of the variables and the direction of the inequality sign when performing operations such as reciprocation.
4. Combine Solutions: Synthesize the solutions obtained from different cases to derive the complete solution set.
5. Verify: It's always beneficial to check the solution set by substituting values from within the range into the original inequality to ensure the inequality holds true.

Real-World Applications

Inequalities are fundamental to numerous real-world applications. They appear in:

• Optimization Problems: Finding the maximum or minimum values of a function subject to constraints is often solved using inequalities.
• Economics: Linear programming, used extensively in resource allocation and production planning, relies heavily on inequalities.
• Engineering: Designing structures and systems often involves considering inequality constraints related to strength, stability, and safety.
• Computer Science: Algorithms and data structures often incorporate inequalities for efficient searching and sorting.

Mastering the skills to solve inequalities opens doors to numerous applications across diverse fields.

Conclusion

Solving inequalities like x - 6/x ≤ x + 7/x - 3 requires a systematic and careful approach. By following the steps outlined in this article, focusing on simplifying the expression, isolating the variable, handling potential undefined cases, and correctly managing the inequality sign during algebraic manipulation, you can confidently solve such problems. Understanding the underlying principles and practicing various examples is key to developing proficiency in this essential mathematical skill. Remember to always verify your solution to ensure accuracy and build a solid understanding of the concepts involved. The more you practice, the more comfortable and proficient you will become at solving these types of inequalities.