To solve the given equation using graphing, we need to follow a series of steps to identify the solutions. The equation presented is:
[tex]\[
\frac{1}{3x + 1} + 1 = \sqrt[5]{5}
\][/tex]
First, let's break down the equation into two separate functions for graphing:
Let [tex]\( f(x) = \frac{1}{3x + 1} + 1 \)[/tex]
Let [tex]\( g(x) = \sqrt[5]{5} \)[/tex]
### Steps:
1. Graph the function [tex]\( f(x) = \frac{1}{3x + 1} + 1 \)[/tex]:
- This function simplifies to [tex]\( f(x) = \frac{1}{3x + 1} + 1 \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\frac{1}{3} \)[/tex], the value of [tex]\( 3x + 1 \)[/tex] approaches 0, making the function undefined (vertical asymptote at [tex]\( x = -\frac{1}{3} \)[/tex]).
- For large positive values of [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( 1 \)[/tex] since [tex]\( \frac{1}{3x + 1} \)[/tex] approaches 0.
- For large negative values of [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] also approaches [tex]\( 1 \)[/tex].
2. Graph the function [tex]\( g(x) = \sqrt[5]{5} \)[/tex]:
- The value of [tex]\( \sqrt[5]{5} \)[/tex] is a constant and can be approximated.
- [tex]\(\sqrt[5]{5} \approx 1.3797\)[/tex], so [tex]\( g(x) = 1.3797 \)[/tex].
3. Find the intersection points of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- To find the solutions to [tex]\(\frac{1}{3x + 1} + 1 = \sqrt[5]{5}\)[/tex], we need to find where the graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect.
Using graphing tools or a graphing calculator, the intersection points of these two functions can be found.
The approximate solutions to the given equation are:
[tex]\[
x \approx -0.761 \quad \text{and} \quad x \approx 1.217
\][/tex]
Thus, the correct answer is:
C. [tex]\( x \approx -0.761 \text{ and } x \approx 1.217 \)[/tex]
