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Using graphing, what are the approximate solutions to this equation?

[tex]\[ \frac{1}{3x+1} + 1 = \sqrt[5]{5} \][/tex]

A. [tex]\( x \approx 28.913 \)[/tex] and [tex]\( x \approx 1.217 \)[/tex]

B. [tex]\( x \approx 0.913 \)[/tex] and [tex]\( x \approx 1.803 \)[/tex]

C. [tex]\( x \approx -0.761 \)[/tex] and [tex]\( x \approx 1.217 \)[/tex]

D. [tex]\( x \approx -0.761 \)[/tex] and [tex]\( x \approx 1.803 \)[/tex]

Sagot :

GinnyAnsweridn
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]
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