1 – e ^(1/x – 1 ) > 0 solve this inequality.........................

To solve the inequality $1-e^{(1/x-1)}>0$, we need to isolate the exponential term and analyze the conditions under which this inequality holds true. Let's break it down step by step.

Step 1: Rearranging the Inequality

First, we can rearrange the inequality:

1 - e^{(1/x - 1)} > 0

implies

e^{(1/x - 1)} < 1.

Step 2: Understanding the Exponential Function

The exponential function $e^y$ is always positive for any real $y$, and it equals 1 when $y=0$. Therefore, we need to find when:

(1/x - 1) < 0.

Step 3: Solving the Exponential Condition

Let’s solve the inequality:

1/x - 1 < 0.

Rearranging gives:

1/x < 1.

Multiplying both sides by $x$ (keeping in mind that this affects the inequality depending on the sign of $x$) yields:

• If $x>0$, then $1<x$ or $x>1$.
• If $x<0$, then $1>x$ (since multiplying by a negative flips the inequality). However, in this case, $1/x$ will be negative, which cannot satisfy the condition $1/x<1$.

Step 4: Analyzing the Results

From our analysis, we conclude that:

1. For $x>0$, the solution is $x>1$.2. For $x<0$, there are no valid solutions as the exponential condition cannot be satisfied.

Final Solution

Thus, the solution to the inequality $1-e^{(1/x-1)}>0$ is:

x > 1.

This means that for any value of $x$ greater than 1, the original inequality holds true. If you have any further questions about specific values or examples, feel free to ask!