Sure, let's solve the equation [tex]\( -(6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-1} \)[/tex] by graphing and rounding to the nearest tenth.
First, let's break down the equation and understand what functions we are dealing with on both sides.
### Step 1: Understand the Equation
1. Left Side of the Equation:
[tex]\(f(x) = -(6)^{x-1} + 5\)[/tex]
2. Right Side of the Equation:
[tex]\(\left(\frac{2}{3}\right)^{2-1}\)[/tex]
Simplify the right side:
[tex]\[
\left(\frac{2}{3}\right)^{2-1} = \left(\frac{2}{3}\right)^1 = \frac{2}{3}
\][/tex]
### Step 2: Rewrite the Equation
Rewriting the original equation with the right side simplified:
[tex]\[
-(6)^{x-1} + 5 = \frac{2}{3}
\][/tex]
### Step 3: Setup for Graphing
We will graph the following two functions:
1. [tex]\( f(x) = -(6)^{x-1} + 5 \)[/tex]
2. [tex]\( g(x) = \frac{2}{3} \)[/tex]
### Step 4: Find the Intersection
We need to find the value of [tex]\( x \)[/tex] where the graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect. This intersection represents the solution to our equation.
The intersection of these two functions gives us the value of [tex]\( x \)[/tex]:
After graphing:
- [tex]\( f(x) = -(6)^{x-1} + 5 \)[/tex]
- [tex]\( g(x) = \frac{2}{3} \)[/tex]
We find that the graphs intersect at approximately [tex]\( x \approx 1.8195 \)[/tex].
### Step 5: Round to the Nearest Tenth
Rounding the value of [tex]\( x \)[/tex] to the nearest tenth:
[tex]\[
x \approx 1.8
\][/tex]
### Conclusion
So, the solution to the equation [tex]\( -(6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-1} \)[/tex] when rounded to the nearest tenth is approximately:
[tex]\[
x \approx 1.8
\][/tex]
Thus, [tex]\( x \approx 1.8 \)[/tex] is the value that satisfies the given equation.
