Select The Correct Answer From The Drop-down Menu.Consider The Equation And The Graph.$\frac{2}{x+4}=3^x+1$

Introduction

When it comes to solving equations with exponents, it's essential to understand the properties of exponents and how to manipulate them to isolate the variable. In this article, we'll explore the equation $\frac{2}{x+4}=3^x+1$ and provide a step-by-step guide on how to solve it.

Understanding the Equation

The given equation is $\frac{2}{x+4}=3^x+1$. This equation involves an exponential term $3^x$ and a rational term $\frac{2}{x+4}$. To solve this equation, we need to isolate the variable $x$.

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction, we can multiply both sides of the equation by the denominator, which is $x+4$. This gives us:

$$2 = (x+4)(3^x+1)$$

Step 2: Expand the Right-Hand Side

Next, we can expand the right-hand side of the equation by multiplying the two binomials:

$$2 = 3^x(x+4) + (x+4)$$

Step 3: Distribute the Exponent

Now, we can distribute the exponent $3^x$ to the terms inside the parentheses:

$$2 = 3^x(x) + 3^x(4) + (x+4)$$

Step 4: Simplify the Equation

We can simplify the equation by combining like terms:

$$2 = x(3^x) + 4(3^x) + x + 4$$

Step 5: Move All Terms to One Side

To isolate the variable $x$, we can move all the terms to one side of the equation:

$$0 = x(3^x) + 4(3^x) + x - 2$$

Step 6: Factor Out the Common Term

We can factor out the common term $x$ from the first two terms:

$$0 = x(3^x + 4) + x - 2$$

Step 7: Factor Out the Common Term Again

We can factor out the common term $x$ again from the first two terms:

$$0 = x(3^x + 4 + 1) - 2$$

Step 8: Simplify the Equation

We can simplify the equation by combining like terms:

$$0 = x(3^x + 5) - 2$$

Step 9: Add 2 to Both Sides

To isolate the term with the exponent, we can add 2 to both sides of the equation:

$$2 = x(3^x + 5)$$

Step 10: Divide Both Sides by the Coefficient

Finally, we can divide both sides of the equation by the coefficient $3^x + 5$:

$$\frac{2}{3^x + 5} = x$$

Conclusion

In this article, we've provided a step-by-step guide on how to solve the equation $\frac{2}{x+4}=3^x+1$. By following these steps, we can isolate the variable $x$ and find the solution to the equation.

Graphing the Equation

To visualize the solution, we can graph the equation $\frac{2}{x+4}=3^x+1$. The graph will show us the values of $x$ that satisfy the equation.

Graphing the Left-Hand Side

The left-hand side of the equation is $\frac{2}{x+4}$. We can graph this function by plotting the points $(x, \frac{2}{x+4})$ for different values of $x$.

Graphing the Right-Hand Side

The right-hand side of the equation is $3^x+1$. We can graph this function by plotting the points $(x, 3^x+1)$ for different values of $x$.

Finding the Intersection

To find the solution to the equation, we need to find the intersection of the two graphs. The intersection point will give us the value of $x$ that satisfies the equation.

Using Technology to Graph the Equation

We can use technology, such as a graphing calculator or a computer algebra system, to graph the equation and find the intersection point.

Graphing the Equation with Technology

Using technology, we can graph the equation $\frac{2}{x+4}=3^x+1$ and find the intersection point.

Finding the Solution

The intersection point of the two graphs gives us the solution to the equation. We can use the graph to estimate the value of $x$ that satisfies the equation.

Conclusion

In this article, we've provided a step-by-step guide on how to solve the equation $\frac{2}{x+4}=3^x+1$. We've also graphed the equation using technology and found the intersection point to estimate the solution.

Final Answer

The final answer is: $\boxed{2}$

Introduction

In our previous article, we provided a step-by-step guide on how to solve the equation $\frac{2}{x+4}=3^x+1$. In this article, we'll answer some common questions that students often have when it comes to solving equations with exponents.

Q: What is the first step in solving an equation with exponents?

A: The first step in solving an equation with exponents is to isolate the variable. This can be done by getting rid of any fractions or decimals that may be present in the equation.

Q: How do I get rid of a fraction in an equation with exponents?

A: To get rid of a fraction in an equation with exponents, you can multiply both sides of the equation by the denominator. This will eliminate the fraction and allow you to work with the equation more easily.

Q: What is the difference between an exponential term and a rational term?

A: An exponential term is a term that involves an exponent, such as $3^x$. A rational term is a term that involves a fraction, such as $\frac{2}{x+4}$.

Q: How do I simplify an equation with exponents?

A: To simplify an equation with exponents, you can combine like terms and eliminate any unnecessary parentheses. You can also use the properties of exponents to simplify the equation.

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: This property is known as the product of powers property. It states that when you multiply two exponential terms with the same base, you can add the exponents.

Q: How do I use the product of powers property to simplify an equation with exponents?

A: To use the product of powers property to simplify an equation with exponents, you can multiply the two exponential terms together and add the exponents.

Q: What is the difference between an exponential equation and a rational equation?

A: An exponential equation is an equation that involves an exponential term, such as $3^x$. A rational equation is an equation that involves a rational term, such as $\frac{2}{x+4}$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the properties of exponents to isolate the variable. You can also use logarithms to solve the equation.

Q: What is the difference between a linear equation and an exponential equation?

A: A linear equation is an equation that involves a linear term, such as $2x$. An exponential equation is an equation that involves an exponential term, such as $3^x$.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use basic algebraic operations, such as addition, subtraction, multiplication, and division, to isolate the variable.

Q: What is the difference between a quadratic equation and an exponential equation?

A: A quadratic equation is an equation that involves a quadratic term, such as $x^2$. An exponential equation is an equation that involves an exponential term, such as $3^x$.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula or factor the equation to find the solutions.

Conclusion

In this article, we've answered some common questions that students often have when it comes to solving equations with exponents. We've also provided a step-by-step guide on how to solve an equation with exponents.

Final Answer

The final answer is: $\boxed{2}$