solve the inequality
e^x+e^x-x=6
can you show work
You don't have an inequality, you have an equation
Why do you have two terms the same?
Is there a typo ?
I am sorry.
This is the equation.
Solve the follow exponential equation
e^x+e^-x=6
e^-x = 1/e^x
so
e^x + 1/e^x = 6
e^2x/e^x + 1/e^x = 6
(e^2x + 1) = 6 e^x
let z = e^x
z^2 + 1 = 6 z
z^2 - 6 z + 1 = 0
z = [ 6 +/- sqrt(36 -4) ] /2
z = 3 +/- .5 sqrt(32)
z = 3 +/- 2 sqrt 2
so
e^x = 5.828 or e^x = .1716
ln e^x = x = 1.76 or x = -1.76
To solve the inequality e^x + e^x - x = 6, we need to find the value of x that makes the equation true.
First, let's simplify the equation:
2e^x - x = 6
To solve for x, we will use a numerical method called the Newton-Raphson method. This method helps us approximate the value of x iteratively.
1. Start by making an initial guess for the value of x. Let's say x = 1.
2. Substitute the guessed value of x into the equation: 2e^1 - 1 = 6
3. Calculate the value of the left side of the equation using the exponential function: 2e - 1 = 6
4. Rearrange the equation to solve for e: 2e = 7
5. Divide both sides by 2: e = 7/2 = 3.5
Now, we have an updated value for e (approximately 3.5). We will use this value in the next iteration.
6. Repeat steps 2-5, using the new value of e, until you find a value that satisfies the equation:
Guess x = 1:
2e^1 - 1 = 6
2e - 1 = 6
2e = 7
e ≈ 7/2 ≈ 3.5
Guess x = 3.5:
2e^3.5 - 3.5 = 6
2(3.5) - 3.5 = 6
7 - 3.5 = 6
3.5 = 6
Since 3.5 is not equal to 6, we need to continue the iteration process until we find a value that satisfies the equation.
7. Repeat steps 6, gradually refining the guess for x, until you reach a satisfactory approximation. The process may take several iterations depending on the desired level of accuracy.
If you prefer a more precise result, you can use software, such as a mathematical calculator or computer program, that supports solving equations numerically. These tools can perform the iteration process much more efficiently and accurately.