Solving Inequalities with Fractions: A Comprehensive Guide

Solving inequalities with fractions can be a challenging task, especially when the problem statement is ambiguous or poorly written. This guide will walk you through the process of solving the given inequality step-by-step and highlight the importance of clear notation. Let us start with the primary problem: ( frac{x^2}{1-x} - 1 geq 0 ).

Understanding the Problem

First, let's clarify the given inequality: ( frac{x^2}{1-x} - 1 geq 0 ). This can be transformed into a more manageable form by adding 1 to both sides. This step is crucial to make the inequality easier to handle.

"  "dfrac{x^2}{1-x} - 1 geq 0
"  "Rightarrow dfrac{x^2 - (1-x)}{1-x} geq 0
"  "Rightarrow dfrac{x^2   x - 1}{1-x} geq 0
"  "

Now, our inequality is simplified to ( frac{x^2 x - 1}{1-x} geq 0 ).

Step-by-Step Solution

Let's solve the inequality ( frac{x^2 x - 1}{1-x} geq 0 ) step-by-step:

Factor the Numerator

First, let's factor the numerator ( x^2 x - 1 ). The roots of the quadratic equation ( x^2 x - 1 0 ) are:

"  "x  frac{-b pm sqrt{b^2 - 4ac}}{2a}
"  "Rightarrow x  frac{-1 pm sqrt{1^2 - 4 cdot 1 cdot (-1)}}{2 cdot 1}
"  "Rightarrow x  frac{-1 pm sqrt{5}}{2}
"  "

So, the roots are ( x frac{-1 sqrt{5}}{2} ) and ( x frac{-1 - sqrt{5}}{2} ).

Analyze the Sign of the Expression

To analyze the sign of the expression ( frac{x^2 x - 1}{1-x} ), we need to determine the intervals where the expression is non-negative. The critical points are the roots of the numerator and the point where the denominator is zero (i.e., ( x 1 )). These points divide the number line into intervals. We test the sign of the expression in each interval:

Interval 1: ( x Interval 2: ( frac{-1 - sqrt{5}}{2} Interval 3: ( 1 Interval 4: ( x > frac{-1 sqrt{5}}{2} )

Determine the Solution Set

By testing the sign of the expression in each interval, we find that the expression ( frac{x^2 x - 1}{1-x} ) is non-negative in the intervals ( left( -infty, frac{-1 - sqrt{5}}{2} right] cup left[ frac{-1 sqrt{5}}{2}, 1 right) cup left( 1, infty right) ).

Common Mistakes to Avoid

One common mistake when solving inequalities with fractions is to clear the denominator. This can change the direction of the inequality if the denominator is negative. To avoid this, we transform the given inequality into a standard form and analyze the sign of the expression.

Conclusion

Solving inequalities with fractions requires careful attention to detail and a systematic approach. By clarifying the problem statement and using appropriate grouping symbols, we can ensure that the solution is accurate and comprehensible.

The key takeaways are:

Transform the inequality into a standard form. Factor the numerator and denominator if possible. Analyze the sign of the expression in each interval. Avoid clearing the denominator to prevent changing the inequality direction.

By following these steps, you can effectively solve inequalities with fractions and ensure accurate results.