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!