Based on the problem and the provided information, let's determine the solution step-by-step:
1. Equation Setup: We are given the equation:
[tex]\[
-(5)^{3-x} + 4 = -2x
\][/tex]
2. Graph Intersection Insight: The given information suggests examining the intersection points of the two functions:
[tex]\[
f(x) = -(5)^{3-x} + 4
\][/tex]
and
[tex]\[
g(x) = -2x
\][/tex]
3. Intersection Points: We need to focus on the points of intersection as they represent the solutions to the equation where [tex]\( f(x) = g(x) \)[/tex].
4. Approximate Solutions:
- The solution is approximately [tex]\(x = 1.7\)[/tex]. This is identified as an intersection point value for [tex]\(x\)[/tex].
- It notes the solution [tex]\( y \approx -3.5 \)[/tex] being an intersection point, but for [tex]\( y \)[/tex]-value. We are mainly interested in the [tex]\(x\)[/tex]-values where [tex]\( f(x) = g(x) \)[/tex].
5. Verification of Solutions:
- Evaluating at approximately [tex]\( x = 1.7 \)[/tex]:
Substituting [tex]\( x \approx 1.7 \)[/tex] into the original equation should satisfy both sides.
- The second value [tex]\( y = -3.5 \)[/tex] does not function as a solution in terms of [tex]\( x \)[/tex], hence we don't consider it in finding solutions.
6. Conclusion:
There is a solution to the equation, and the approximate value of the solution is:
[tex]\[
x \approx 1.7
\][/tex]
So, the single approximate solution for the equation [tex]\(- (5)^{3 - x} + 4 = -2x\)[/tex] is indeed approximately [tex]\( x = 1.7 \)[/tex], given it corresponds to the intersection point of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
