Solve for x?
xln(x) + 1 > x + ln(x)
e^1/x > 1
Any help would be appreciated. THANK YOU!
(2) e^(1/x) > 1
When e^(1/x) = 1, (taking natural logs of both sides)
(1/x) = 0
x = infinity
Any positive number x that is less than infinity satisfies the inequality.
(1) The solution to
xln(x) + 1 = x + ln(x)
is x = 1. Values x between 0 and satisfy the inequality.
To solve the inequality xln(x) + 1 > x + ln(x), you can follow these steps:
Step 1: Rearrange the inequality to isolate the terms involving x on one side:
xln(x) - x - ln(x) + 1 > 0
Step 2: Combine the like terms:
xln(x) - ln(x) - x + 1 > 0
Step 3: Simplify:
ln(x)(x - 1) - (x - 1) > 0
Step 4: Factor out the common term:
(x - 1)(ln(x) - 1) > 0
Step 5: Determine the critical points by setting each factor equal to zero:
x - 1 = 0 (1)
ln(x) - 1 = 0 (2)
Solve equation (1):
x = 1
Solve equation (2):
ln(x) = 1
x = e
So, the critical points are x = 1 and x = e.
Step 6: Create a number line and plot the critical points on it:
e 1
Step 7: Test the intervals between the critical points:
Choose a value less than 1, let's say x = 0. Substitute it into the inequality:
(0 - 1)(ln(0) - 1) > 0
(-1)(undefined) > 0
Since ln(0) is undefined, we cannot evaluate whether it is greater than 0 or not.
Choose a value between 1 and e, let's say x = 2. Substitute it into the inequality:
(2 - 1)(ln(2) - 1) > 0
(1)(positive) > 0
Since the inequality is true, this interval satisfies the inequality.
Choose a value greater than e, let's say x = 3. Substitute it into the inequality:
(3 - 1)(ln(3) - 1) > 0
(2)(positive) > 0
Since the inequality is true, this interval satisfies the inequality.
Step 8: Determine the solution:
From the number line, we see that the solution to the inequality is x ∈ (1, e) U (e, ∞).
Now, let's move on to solving the inequality e^(1/x) > 1:
Step 1: Raise e to both sides of the inequality:
e^(1/x) > e^1
Step 2: Simplify:
e^(1/x) > e
Step 3: Take the natural logarithm of both sides:
ln(e^(1/x)) > ln(e)
Step 4: Simplify using the properties of logarithms:
1/x > 1
Step 5: Cross-multiply and simplify:
1 > x
So, the solution to the inequality e^(1/x) > 1 is x < 1.
Therefore, combining both solutions, the final solution is x < 1.