Solve [tex]$-(6)^{x-1}+5=\left(\frac{2}{3}\right)^{2-1}$[/tex] by graphing. Round to the nearest tenth.

[tex]$x \approx 1$[/tex]



Answer :

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.