Solving an Inequality: 1/x < 1/(x-1)
Stuck on inequalities? This guide will walk you through solving the inequality: \( \frac{1}{x} < \frac{1}{x-1} \).
We’ll solve the inequality step-by-step, combining the fractions, simplifying the expression, and then using a number line to find the solution set. Let’s dive in!
By following these steps and analyzing the number line, we can determine that the solution set for the inequality is \( \{x \mid x < 0 \text{ or } x > 1\} \).
Identifying Functions 1/x and 1/(x-1) in Graph
The provided image features two curves on a coordinate plane, which represent the functions 1/x and 1/(x-1). The 1/x function is represented by the black curve, while the 1/(x-1) function is represented by the green curve.
In the regions where x is less than 0, the black curve is positioned below the green curve. This indicates that for x values less than 0, the value of the function 1/x is less than the value of the function 1/(x-1).
For x values greater than 1, the black curve remains below the green curve. This confirms that in this region, the function 1/x is also less than the function 1/(x-1).
The graph visually supports the solution to the inequality, showing that for x less than 0 and x greater than 1, the function 1/x is less than 1/(x-1).
