Solve the equation for $x$ by graphing.
$-4 x-1=5^n+4$
A. $x \approx-0.25$
B. $x \approx-1.50$
C. $x \approx-1.25$
D. $x \approx-1
Answer (2)
Isolate x in the equation: x = − 4 5 n + 5 .
Test the given options for x and solve for n .
Option B, x ≈ − 1.50 , gives a valid solution n = 0 .
Therefore, the correct answer is x ≈ − 1.50 .
Explanation
Understanding the Problem We are given the equation − 4 x − 1 = 5 n + 4 and asked to solve for x by graphing. The value of n is not specified, which means x will be expressed in terms of n . We need to find the value of x that satisfies the equation for some value of n . We can test the given options to see which one satisfies the equation for a reasonable value of n .
Isolating x First, let's isolate x in the equation: − 4 x − 1 = 5 n + 4
Add 1 to both sides: − 4 x = 5 n + 5
Divide both sides by -4: x = − 4 5 n + 5
Testing the Options Now, let's test the given options for x and see if we can find a corresponding value for n :
A. x ≈ − 0.25 − 0.25 = − 4 5 n + 5
1 = 5 n + 5
5 n = − 4
This is not possible since 5 n is always positive for any real n .
B. x ≈ − 1.50 − 1.50 = − 4 5 n + 5
6 = 5 n + 5
5 n = 1
n = 0
This is a valid solution. When n = 0 , x = − 1.5 .
C. x ≈ − 1.25 − 1.25 = − 4 5 n + 5
5 = 5 n + 5
5 n = 0
This is not possible since 5 n is always positive for any real n .
D. x ≈ − 1 − 1 = − 4 5 n + 5
4 = 5 n + 5
5 n = − 1
This is not possible since 5 n is always positive for any real n .
Conclusion From the above analysis, only option B gives a valid solution for n . When x = − 1.5 , we have n = 0 . Therefore, the correct answer is x ≈ − 1.50 .
Examples
Consider a scenario where you are analyzing the decay of a radioactive substance. The amount of substance remaining after time t can be modeled by an exponential equation. If you know the amount remaining at a certain time and want to find the decay constant, you would solve a similar equation involving exponentials. This type of problem is also applicable in financial models, such as calculating the depreciation of an asset over time or determining the growth rate of an investment.
The solution to the equation − 4 x − 1 = 5 n + 4 when tested against the provided options shows that only x ≈ − 1.50 is valid, which corresponds to n = 0 . Therefore, the correct choice is option B. This can be confirmed by substituting x = − 1.50 back into the equation.
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