Select the correct answer.

Using a table of values, determine the solution to the equation below to the nearest fourth of a unit.

[tex]$2 x-4=3^{-x}+1$

A. [tex]$x \approx 3.25$
B. [tex]$x \approx 2.5$
C. [tex]$x \approx 1$
D. [tex]$x \approx 2$

Rewrite the equation as f ( x ) = 2 x − 5 − 3 − x = 0 .
Evaluate f ( x ) for each option: x = 3.25 , 2.5 , 1 , 2 .
Find the value of x where ∣ f ( x ) ∣ is the smallest.
The solution is x ≈ 2.5 because ∣ f ( 2.5 ) ∣ ≈ 0.064 is the smallest. 2.5 ​

Explanation

Problem Analysis We are given the equation 2 x − 4 = 3 − x + 1 and asked to find the solution to the nearest fourth of a unit using a table of values. The options are A. x ≈ 3.25 , B. x ≈ 2.5 , C. x ≈ 1 , D. x ≈ 2 .

Rewriting the Equation First, let's rewrite the equation as 2 x − 4 − 3 − x − 1 = 0 , which simplifies to 2 x − 5 − 3 − x = 0 . We can define a function f ( x ) = 2 x − 5 − 3 − x . Our goal is to find the value of x for which f ( x ) is closest to 0.

Evaluating the Function Now, we will evaluate f ( x ) for each of the given values of x in the options: x = 3.25 , 2.5 , 1 , 2 .

Calculating f(3.25) For x = 3.25 , we have f ( 3.25 ) = 2 ( 3.25 ) − 5 − 3 − 3.25 = 6.5 − 5 − 3 − 3.25 = 1.5 − 3 − 3.25 . Since 3 − 3.25 ≈ 0.028 , f ( 3.25 ) ≈ 1.5 − 0.028 = 1.472 .

Calculating f(2.5) For x = 2.5 , we have f ( 2.5 ) = 2 ( 2.5 ) − 5 − 3 − 2.5 = 5 − 5 − 3 − 2.5 = − 3 − 2.5 . Since 3 − 2.5 ≈ 0.064 , f ( 2.5 ) ≈ − 0.064 .

Calculating f(1) For x = 1 , we have f ( 1 ) = 2 ( 1 ) − 5 − 3 − 1 = 2 − 5 − 3 1 ​ = − 3 − 3 1 ​ = − 3 10 ​ ≈ − 3.333 .

Calculating f(2) For x = 2 , we have f ( 2 ) = 2 ( 2 ) − 5 − 3 − 2 = 4 − 5 − 9 1 ​ = − 1 − 9 1 ​ = − 9 10 ​ ≈ − 1.111 .

Finding the Smallest Absolute Value We want to find the value of x for which ∣ f ( x ) ∣ is the smallest. We have ∣ f ( 3.25 ) ∣ ≈ 1.472 , ∣ f ( 2.5 ) ∣ ≈ 0.064 , ∣ f ( 1 ) ∣ ≈ 3.333 , and ∣ f ( 2 ) ∣ ≈ 1.111 . The smallest absolute value is ∣ f ( 2.5 ) ∣ ≈ 0.064 .

Conclusion Therefore, the solution to the equation 2 x − 4 = 3 − x + 1 to the nearest fourth of a unit is x ≈ 2.5 .


Examples
Imagine you're trying to find the break-even point for a new business venture. The equation 2 x − 4 = 3 − x + 1 can be analogous to a cost-benefit scenario, where x represents a certain level of production or sales. The left side, 2 x − 4 , could represent the profit margin increasing linearly with each sale, while the right side, 3 − x + 1 , could represent diminishing returns or decreasing costs as production increases. Solving this equation helps you determine the exact point where your profits balance out your costs, guiding your business decisions to optimize for success. This type of problem demonstrates how mathematical modeling can provide valuable insights into real-world scenarios, aiding in strategic planning and decision-making.

Answered by GinnyAnswer | 2025-07-03

The solution to the equation 2 x − 4 = 3 − x + 1 is found by evaluating a function derived from the equation. The closest value of x that minimizes the function is approximately 2.5 . Thus, the correct answer is option B: x ≈ 2.5 .
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Answered by Anonymous | 2025-07-04