Use a graphing tool to solve the equation for $x$.
$-4 x-1=5^x+4$
A. $x \approx-0.25$
B. $x \approx-1.25$
C. $x \approx-1$
D. $x \approx-1.50$
Answer (2)
Rewrite the equation as 5 x + 4 x + 5 = 0 .
Find the root of the equation 5 x + 4 x + 5 = 0 .
The approximate root is x ≈ − 1.28177 .
The closest option is x ≈ − 1.25 , so the final answer is x ≈ − 1.25 .
Explanation
Understanding the Problem We are asked to solve the equation − 4 x − 1 = 5 x + 4 for x using a graphing tool and to select the correct approximation from the given options.
Rewriting the Equation First, let's rewrite the equation as 5 x + 4 x + 5 = 0 . This form is useful for finding the roots of the function f ( x ) = 5 x + 4 x + 5 .
Finding the Root Now, we need to find the value of x for which f ( x ) = 0 . This can be done using a graphing tool or numerical methods. Using a numerical method, we find that x ≈ − 1.28177 .
Selecting the Correct Option Comparing this value with the given options:
A. x ≈ − 0.25
B. x ≈ − 1.25
C. x ≈ − 1
D. x ≈ − 1.50 The closest approximation to our calculated value is x ≈ − 1.25 .
Examples
Understanding exponential equations like − 4 x − 1 = 5 x + 4 is crucial in many real-world applications, such as modeling population growth or radioactive decay. For example, if you're studying the decay of a radioactive substance, you might use a similar equation to determine how long it takes for the substance to reach a safe level. The variable x could represent time, and the equation helps predict the amount of substance remaining after a certain period. Solving these equations accurately allows for precise predictions and informed decisions in various scientific and engineering fields.
To solve the equation − 4 x − 1 = 5 x + 4 , we rewrite it to find f ( x ) = 5 x + 4 x + 5 = 0 . After graphing or using numerical methods, we find the approximate root is x ≈ − 1.28177 , leading us to select option B: x ≈ − 1.25 .
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