Use the given graph to determine if there are any solutions to the equation [tex]\( -(5)^{3-x} + 4 = -2x \)[/tex].

A. The solution is approximately 1.7 because it is the [tex]\( x \)[/tex]-value of the intersection of the functions.
B. The solution is approximately -3.5 because it is the [tex]\( y \)[/tex]-value of the intersection of the functions.
C. There is no solution to the equation.
D. The two solutions to the equation are 1.7 and -3.5 because that is the intersection of the functions.



Answer :

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].