Solving Inequalities Involving Fractions: A Comprehensive Guide

When working with inequalities that involve fractions, it is important to follow a systematic approach to ensure accuracy. This guide will walk you through the detailed process using the example of the inequality x1 / x2 5. By the end of this article, you will understand the step-by-step solution and be able to apply similar methods to other fraction-based inequalities. Let's dive into the solution process.

Solving the Inequality x1 / x2 5

Given the inequality: (frac{x_1}{x_2} 5)

Step 1: Express the Inequality as a Quadratic Equation

First, consider the equation: (frac{x_1}{x_2} 5). This equation can be rewritten as a quadratic equation to better understand its behavior:

[x_1 - 5x_2 0]

From here, we can solve for x1:

[x_1 5x_2]

Step 2: Analyze and Simplify

To find the solution to the inequality (frac{x_1}{x_2} 5), we need to analyze the inequality under different cases:

Case 1: x2 0

In this case, we rewrite the inequality:

[frac{x_1}{x_2} 5]

By multiplying both sides by x2 (since it is positive, the inequality sign remains the same):

[x_1 5x_2]

Case 2: x2 0

In this case, multiplying both sides by x2 reverses the inequality sign:

[x_1 5x_2]

Combining the Results

For Case 1, we have:

[x_1 5x_2] with [x_2 0]

For Case 2, we have:

[x_1 5x_2] with [x_2 0]

Combining these results, the solution can be expressed as:

[x otin left[-frac{9}{4} leq x leq -frac{11}{6}right]]

General Approach for Solving Inequalities

In general, to solve the inequality (f(x) a), follow these steps:

Step 1: Solve (f(x) a)

Solve the equation (f(x) a) to find the boundary points. These points will help determine the intervals to test.

Step 2: Solve (f(x) a) and (f(x) a)

Consider the inequality separately for both (f(x) a) and (f(x) a) to find the intervals where the inequality holds true. Combine these intervals to get the overall solution.

Example: Solving (x frac{1}{x} - 2 5)

Step 1: Express as an Equation

To solve (x frac{1}{x} - 2 5), first set the equation to zero:

[x frac{1}{x} - 2 5] or [x frac{1}{x} - 2 -5]

Solving (x frac{1}{x} - 2 5)

[x^2 - 7x 1 0]

Solving the quadratic equation:

[x frac{7 pm sqrt{49 - 4}}{2} frac{7 pm sqrt{45}}{2} frac{7 pm 3sqrt{5}}{2} approx 6.854, 0.146]

Solving (x frac{1}{x} - 2 -5)

[x^2 3x 1 0]

Solving the quadratic equation:

[x frac{-3 pm sqrt{9 - 4}}{2} frac{-3 pm sqrt{5}}{2} approx -3.618, -0.382]

Step 2: Check the Intervals

Determine the intervals by checking points between and outside the boundary points. For (x frac{1}{x} - 2 5), the boundary points are (x approx -3.618, -0.382, 0.146, 6.854).

Checking points in each interval:

-3: true -1: false -0.1: false 4: true 7: false

The solution is:

[x in left[-frac{3 sqrt{5}}{2}, frac{-3 - sqrt{5}}{2}right] cup left[frac{7 - 3sqrt{5}}{2}, frac{7 3sqrt{5}}{2}right]]

Solving (frac{x_1}{x-2} 5)

If the original inequality is (frac{x_1}{x-2} 5), the process is similar:

Step 1: Set the Equation to Zero

[frac{x_1}{x-2} - 5 0] and [frac{x_1}{x-2} 5 0]

Solving (frac{x_1}{x-2} - 5 0)

[x_1 - 5(x - 2) 0 Rightarrow x_1 - 5x 10 0 Rightarrow x frac{10}{x_1 - 5}]

Solving (frac{x_1}{x-2} 5 0)

[x_1 5(x - 2) 0 Rightarrow x_1 5x - 10 0 Rightarrow x frac{10 - x_1}{5}]

Boundary points: (x 2, frac{10}{x_1 - 5}, frac{10 - x_1}{5})

Checking points between and outside these boundary points:

0: true 1.6: false 2.1: false 3: true

The solution is:

[x in (-infty, frac{3}{2}) cup (frac{11}{4}, infty)]

Conclusion

Understanding and applying these steps will help you solve similar fraction-based inequalities. By analyzing the inequality, solving the related equations, and testing intervals, you can accurately determine the solution set. Practice with different problems to enhance your skills in solving inequalities involving fractions.