To solve the equation [tex]\( x + 5 = -3^x + 4 \)[/tex] to the nearest fourth of a unit, we need to find the value of [tex]\( x \)[/tex] that satisfies this equation. Here’s a step-by-step outline to arrive at the solution:
Step 1: Understand the Equation
The given equation is:
[tex]\[ x + 5 = -3^x + 4 \][/tex]
We want to find the value of [tex]\( x \)[/tex] such that both sides are equal.
Step 2: Rearrange the Equation
To isolate [tex]\( x \)[/tex], we can rearrange the equation:
[tex]\[ x + 5 - 4 = -3^x \][/tex]
[tex]\[ x + 1 = -3^x \][/tex]
Step 3: Create a Table of Values
Let’s evaluate both functions, [tex]\( f(x) = x + 1 \)[/tex] and [tex]\( g(x) = -3^x \)[/tex], at several points to see where they might intersect. Here are some sample values:
| [tex]\( x \)[/tex] | [tex]\( f(x) = x + 1 \)[/tex] | [tex]\( g(x) = -3^x \)[/tex] |
|:--------:|:--------------------:|:------------------:|
| -2.0 | -1.0 | -0.111 |
| -1.5 | -0.5 | -0.197 |
| -1.25 | -0.25 | -0.316 |
| -1.0 | 0.0 | -1.0 |
| 0.0 | 1.0 | -1.0 |
| 0.5 | 1.5 | -0.593 |
| 1.0 | 2.0 | -3.0 |
Notice that around [tex]\( x = -1.25 \)[/tex], the values [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are close to each other. This suggests that [tex]\( x = -1.25 \)[/tex] is a potential solution.
Step 4: Verify the Approximation
To confirm that [tex]\( x = -1.25 \)[/tex] is indeed the solution, let’s substitute back into the original equations:
[tex]\[ f(-1.25) = -1.25 + 1 = -0.25 \][/tex]
[tex]\[ g(-1.25) = -3^{-1.25} \approx -0.316 \][/tex]
The results are very close, affirming that the intersection point is roughly [tex]\( x \approx -1.25 \)[/tex].
Step 5: Conclusion
The solution to the equation [tex]\( x + 5 = -3^x + 4 \)[/tex] to the nearest fourth of a unit is [tex]\( x \approx -1.25 \)[/tex].
Correct Answer:
A. [tex]\( x \approx -1.25 \)[/tex]
