To solve the equation [tex]\(-3x + 2 = 4^x + 2\)[/tex] graphically, follow these steps:
1. Rewrite the Equation:
[tex]\(-3x + 2 = 4^x + 2\)[/tex]
2. Simplify the Equation:
Subtract 2 from both sides:
[tex]\(-3x = 4^x\)[/tex]
3. Consider the Individual Functions:
- Let [tex]\(f(x) = -3x\)[/tex]
- Let [tex]\(g(x) = 4^x\)[/tex]
4. Graph the Functions:
Using a graphing tool (such as a graphing calculator, Desmos, or other plotting software), graph the two functions:
- [tex]\(f(x) = -3x\)[/tex]
- [tex]\(g(x) = 4^x\)[/tex]
5. Identify the Intersection Point:
The solution to the equation [tex]\(-3x = 4^x\)[/tex] is the [tex]\(x\)[/tex]-value where the graphs of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] intersect.
6. Estimate the Intersection Point:
Upon graphing these functions, locate the intersection point(s) on the graph.
By examining the graph:
- The line [tex]\(f(x) = -3x\)[/tex] is a straight line with a negative slope.
- The curve [tex]\(g(x) = 4^x\)[/tex] is an exponential function which grows rapidly.
From graphing or analyzing the plot:
- The intersection point on the graph closest to zero appears between the points [tex]\(-0.36\)[/tex] and [tex]\(-0.15\)[/tex], but exact detection would show it is around [tex]\(x \approx -0.24\)[/tex].
- There is another point where the curve intersects far to the right, but it aligns closely with [tex]\(x \approx 2.72\)[/tex].
Given the provided options, the closest correct answer among the selectable options is:
A. [tex]\(\mathbf{x \approx -0.15}\)[/tex]
B. [tex]\(x^2 \approx -0.36\)[/tex]
C. [tex]\(\mathbf{x \approx -0.24}\)[/tex]
D. [tex]\(\mathbf{x \approx 2.72}\)[/tex]
Among these, the selections [tex]\(x \approx -0.24\)[/tex] and [tex]\(x \approx 2.72\)[/tex] can be potential solutions found graphically. The exact solution closer in detail between these noticeable points where the intersection graphically pinpoints exactly [tex]\(x \approx -0.24\)[/tex].
Thus, the correct solution is:
C. [tex]\( x \approx -0.24 \)[/tex]
