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

sadaf sagheer , 6 Years ago
Grade 12th pass
anser 4 Answers
Saurabh Koranglekar

Last Activity: 6 Years ago

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!

Aditya Gupta

Last Activity: 6 Years ago

note that the above answer by saurabh is wrong.
we see that e ^(1/x – 1 ) 1
1/x – 1
or (x – 1)/x > 0
or x belongs to (–infinity, 0)U(1, infinity)

Aditya Gupta

Last Activity: 6 Years ago

we see that e ^(1/x – 1 ) is less than 1
taking log base e both sides 
or x belongs to (–infinity, 0)U(1, infinity)(x – 1)/x is greater than 0is less than 0(1/x) – 1 

Aditya Gupta

Last Activity: 6 Years ago

due to technical error the it has become impossible to write the solution. but all you gotta do is to take log base e both sides and then solve using wavy curve method to get the final CORRECT answer as:
x belongs to :
(–infinity, 0)U(1, infinity)

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