Final answer:
To approximate the solution to the equation using three iterations of successive approximation, we can use the iterative method. However, after three iterations, we have not found a solution to the equation. Therefore, we cannot approximate the value of x using three iterations.
Step-by-step explanation:
To approximate the solution to the equation -(3/2)^x + 12 = 2x - 3 using three iterations of successive approximation, we can use the iterative method. We start with an initial guess for the value of x, and then use the equation to update the guess in each iteration.
Let's start with an initial guess of x = 0. Substitute this value into the equation to get a new value for x:
-(3/2)^0 + 12 = 2(0) - 3
Simplifying the equation, we get:
1 + 12 = -3
This is not a true statement, so our initial guess of x = 0 is not the solution. Let's move on to the first iteration.
Using the same equation, substitute the updated guess value of x into the equation:
-(3/2)^0.00139 + 12 = 2(0.00139) - 3
Simplifying the equation, we get:
11.886 = 0.00278 - 3
This is not a true statement, so our updated guess of x = 0.00139 is not the solution. Let's move on to the second iteration.
Using the same equation, substitute the updated guess value of x into the equation:
-(3/2)^0.00278 + 12 = 2(0.00278) - 3
Simplifying the equation, we get:
11.5009 = 0.00556 - 3
This is not a true statement, so our updated guess of x = 0.00278 is not the solution. Let's move on to the third iteration.
Using the same equation, substitute the updated guess value of x into the equation:
-(3/2)^0.00556 + 12 = 2(0.00556) - 3
Simplifying the equation, we get:
11.0227 = 0.01112 - 3
This is not a true statement, so our updated guess of x = 0.00556 is not the solution. After three iterations of successive approximation, we have not found a solution to the equation. Therefore, we cannot approximate the value of x using three iterations.