Solving the Rational Inequality x - 3/x ≤ 0
Understanding and solving rational inequalities, such as the one presented in the problem x - 3/x ≤ 0, is a crucial skill in algebra. Rational inequalities often require a systematic approach to identify the intervals where the inequality holds true. In this article, we will explore the detailed steps to solve such inequalities, including using inequalities, algebraic manipulation, and analyzing the critical points.
Problem Restatement
The problem given is x - 3/x ≤ 0. This can be interpreted as two possible expressions based on the notation: x - 3/x or x - (3/x). We will solve the latter, which is the more common interpretation in mathematical contexts.
Step-by-Step Solution
1. Rewrite the Inequality
The inequality can be rewritten as:
x - 3/x ≤ 0
Adding 3/x to both sides gives:
x ≤ 3/x
2. Consider the Critical Points
The critical points are the values of x that make the expression equal to zero (numerator) or undefined (denominator). For the expression 3/x, the critical points are x 0 and x 3.
3. Analyze the Sign of the Expression
We need to analyze the sign of the expression x - 3/x in the different intervals determined by the critical points.
When x > 0:
For x > 0, both x and 3/x are positive. We can multiply both sides by x (which is positive) without changing the inequality:
x 2 ≤ 3
This gives:
x ≤ √3
Therefore, in the interval 0 x ≤ √3, the inequality holds.
When x
For x
x sup2; ≥ 3
Which gives:
x ≤ -√3
Therefore, in the interval -∞ x ≤ -√3, the inequality holds.
4. Combine the Intervals
The solution to the inequality is the union of the intervals where the inequality is satisfied:
x ∈ (-∞, -√3] ∪ [0, √3]
Conclusion
Understanding how to solve rational inequalities like x - 3/x ≤ 0 is essential for advancing your algebraic skills. Each step, from rewriting the inequality to analyzing the critical points, is crucial in finding the correct solution. By mastering these techniques, you will be better equipped to handle more complex algebraic problems and inequalities in various fields, including mathematics, physics, and engineering.
Additional Tips
1. Always check the critical points: Critical points help identify where the inequality changes direction. In this case, x 0 and x 3 are critical points.
2. Use algebraic manipulation carefully: When multiplying both sides of an inequality by a term, pay attention to whether that term is positive or negative. This will determine whether the inequality sign changes.
3. Test intervals: After identifying the critical points, test intervals around them to ensure the inequality holds true. This helps verify your solution.