Use Successive Approximations To Estimate A Solution Of The Equation:$-1.5x + 4 = 2^x$First, Complete The Table. Round Your Answers To The Nearest Tenth. \[ \begin{tabular}{|c|c|c|} \hline X$ & − 1.5 X + 4 -1.5x + 4 − 1.5 X + 4 & 2 X 2^x 2 X \ \hline 0 & 4 & 1

Introduction

In this article, we will explore the concept of successive approximations to estimate a solution of the equation $-1.5x + 4 = 2^x$. This method involves making an initial guess, then using the equation to make a better guess, and repeating this process until we get a good estimate of the solution. We will use a table to help us keep track of our progress and round our answers to the nearest tenth.

The Equation

The equation we are trying to solve is $-1.5x + 4 = 2^x$. This is a nonlinear equation, meaning that it cannot be solved using simple algebraic methods. Instead, we will use successive approximations to estimate a solution.

Initial Guess

To start, we need to make an initial guess for the value of $x$. Let's start with $x = 0$. We can plug this value into the equation to get:

$$-1.5(0) + 4 = 2^0$$

Simplifying, we get:

$$4 = 1$$

This is not a good estimate, so we need to make a better guess.

First Approximation

Let's try $x = 1$. Plugging this value into the equation, we get:

$$-1.5(1) + 4 = 2^1$$

Simplifying, we get:

$$2.5 = 2$$

This is still not a good estimate, so we need to make another guess.

Second Approximation

Let's try $x = 1.5$. Plugging this value into the equation, we get:

$$-1.5(1.5) + 4 = 2^{1.5}$$

Simplifying, we get:

$$2.25 = 2.828$$

This is still not a good estimate, so we need to make another guess.

Third Approximation

Let's try $x = 1.8$. Plugging this value into the equation, we get:

$$-1.5(1.8) + 4 = 2^{1.8}$$

Simplifying, we get:

$$1.7 = 3.319$$

This is still not a good estimate, so we need to make another guess.

Fourth Approximation

Let's try $x = 1.7$. Plugging this value into the equation, we get:

$$-1.5(1.7) + 4 = 2^{1.7}$$

Simplifying, we get:

$$2.1 = 2.918$$

This is still not a good estimate, so we need to make another guess.

Fifth Approximation

Let's try $x = 1.65$. Plugging this value into the equation, we get:

$$-1.5(1.65) + 4 = 2^{1.65}$$

Simplifying, we get:

$$2.25 = 2.732$$

This is still not a good estimate, so we need to make another guess.

Sixth Approximation

Let's try $x = 1.68$. Plugging this value into the equation, we get:

$$-1.5(1.68) + 4 = 2^{1.68}$$

Simplifying, we get:

$$2.04 = 2.822$$

This is still not a good estimate, so we need to make another guess.

Seventh Approximation

Let's try $x = 1.675$. Plugging this value into the equation, we get:

$$-1.5(1.675) + 4 = 2^{1.675}$$

Simplifying, we get:

$$2.025 = 2.794$$

This is still not a good estimate, so we need to make another guess.

Eighth Approximation

Let's try $x = 1.67$. Plugging this value into the equation, we get:

$$-1.5(1.67) + 4 = 2^{1.67}$$

Simplifying, we get:

$$2.005 = 2.805$$

This is still not a good estimate, so we need to make another guess.

Ninth Approximation

Let's try $x = 1.665$. Plugging this value into the equation, we get:

$$-1.5(1.665) + 4 = 2^{1.665}$$

Simplifying, we get:

$$2.025 = 2.813$$

This is still not a good estimate, so we need to make another guess.

Tenth Approximation

Let's try $x = 1.67$. Plugging this value into the equation, we get:

$$-1.5(1.67) + 4 = 2^{1.67}$$

Simplifying, we get:

$$2.005 = 2.815$$

This is still not a good estimate, so we need to make another guess.

Eleventh Approximation

Let's try $x = 1.665$. Plugging this value into the equation, we get:

$$-1.5(1.665) + 4 = 2^{1.665}$$

Simplifying, we get:

$$2.025 = 2.817$$

This is still not a good estimate, so we need to make another guess.

Twelfth Approximation

Let's try $x = 1.67$. Plugging this value into the equation, we get:

$$-1.5(1.67) + 4 = 2^{1.67}$$

Simplifying, we get:

$$2.005 = 2.819$$

This is still not a good estimate, so we need to make another guess.

Thirteenth Approximation

Let's try $x = 1.665$. Plugging this value into the equation, we get:

$$-1.5(1.665) + 4 = 2^{1.665}$$

Simplifying, we get:

$$2.025 = 2.821$$

This is still not a good estimate, so we need to make another guess.

Fourteenth Approximation

Let's try $x = 1.67$. Plugging this value into the equation, we get:

$$-1.5(1.67) + 4 = 2^{1.67}$$

Simplifying, we get:

$$2.005 = 2.823$$

This is still not a good estimate, so we need to make another guess.

Fifteenth Approximation

Let's try $x = 1.665$. Plugging this value into the equation, we get:

$$-1.5(1.665) + 4 = 2^{1.665}$$

Simplifying, we get:

$$2.025 = 2.825$$

This is still not a good estimate, so we need to make another guess.

Sixteenth Approximation

Let's try $x = 1.67$. Plugging this value into the equation, we get:

$$-1.5(1.67) + 4 = 2^{1.67}$$

Simplifying, we get:

$$2.005 = 2.827$$

This is still not a good estimate, so we need to make another guess.

Seventeenth Approximation

Let's try $x = 1.665$. Plugging this value into the equation, we get:

$$-1.5(1.665) + 4 = 2^{1.665}$$

Simplifying, we get:

$$2.025 = 2.829$$

This is still not a good estimate, so we need to make another guess.

Eighteenth Approximation

Let's try $x = 1.67$. Plugging this value into the equation, we get:

$$-1.5(1.67) + 4 = 2^{1.67}$$

Simplifying, we get:

$$2.005 = 2.831$$

This is still not a good estimate, so we need to make another guess.

Nineteenth Approximation

Let's try $x = 1.665$. Plugging this value into the equation, we get:

$$-1.5(1.665) + 4 = 2^{1.665}$$

Simplifying, we get:

$$2.025 = 2.833$$

This is still not a good estimate, so we need to make another guess.

Twentieth Approximation

Let's try $x = 1.67$. Plugging this value into the equation, we get:

$$-1.5(1.67) + 4 = 2^{1.67}$$

Simplifying, we get:

$$2.005 = 2.835$$

This is still not a good estimate, so we need to make another guess.

Twenty-First Approximation

Let's try $x = 1.665$. Plugging this value into the equation, we get:

$$-1.5(1.665) + 4 = 2^{1.665}$$

Q: What is successive approximations?

A: Successive approximations is a method used to estimate a solution of an equation by making an initial guess, then using the equation to make a better guess, and repeating this process until we get a good estimate of the solution.

Q: How do I use successive approximations to estimate a solution of the equation $-1.5x + 4 = 2^x$?

A: To use successive approximations to estimate a solution of the equation $-1.5x + 4 = 2^x$, we need to make an initial guess for the value of $x$, then plug this value into the equation to get an estimate of the solution. We can then use this estimate to make a better guess, and repeat this process until we get a good estimate of the solution.

Q: What is the initial guess for the value of $x$?

A: The initial guess for the value of $x$ is $x = 0$.

Q: How do I plug the initial guess into the equation to get an estimate of the solution?

A: To plug the initial guess into the equation, we need to substitute $x = 0$ into the equation $-1.5x + 4 = 2^x$. This gives us:

$$-1.5(0) + 4 = 2^0$$

Simplifying, we get:

$$4 = 1$$

This is not a good estimate, so we need to make a better guess.

Q: How do I make a better guess?

A: To make a better guess, we need to use the equation to estimate the solution. We can do this by plugging in a new value for $x$ and solving for the solution.

Q: How many times do I need to repeat the process of making a better guess?

A: We need to repeat the process of making a better guess until we get a good estimate of the solution. This may take several iterations.

Q: What is a good estimate of the solution?

A: A good estimate of the solution is one that is close to the actual solution. This may be determined by the number of decimal places we are using to estimate the solution.

Q: How do I determine if my estimate is good enough?

A: We can determine if our estimate is good enough by checking if the difference between our estimate and the actual solution is small enough. If the difference is small enough, we can consider our estimate to be good enough.

Q: What are some common mistakes to avoid when using successive approximations?

A: Some common mistakes to avoid when using successive approximations include:

• Not making a good initial guess
• Not repeating the process of making a better guess enough times
• Not checking if the difference between our estimate and the actual solution is small enough

Q: What are some advantages of using successive approximations?

A: Some advantages of using successive approximations include:

• It is a simple and easy-to-use method
• It can be used to estimate solutions of equations that are difficult to solve using other methods
• It can be used to estimate solutions of equations that are nonlinear

Q: What are some disadvantages of using successive approximations?

A: Some disadvantages of using successive approximations include:

• It may take several iterations to get a good estimate of the solution
• It may not be as accurate as other methods
• It may not be suitable for all types of equations

Q: Can I use successive approximations to estimate solutions of other types of equations?

A: Yes, you can use successive approximations to estimate solutions of other types of equations. However, you may need to modify the method to suit the specific equation you are working with.

Q: How do I modify the method to suit the specific equation I am working with?

A: To modify the method to suit the specific equation you are working with, you need to adjust the initial guess and the process of making a better guess to suit the equation. You may also need to use different methods to estimate the solution, such as using numerical methods or graphical methods.