Select the correct answer. Using a table of values, determine the solution to the equation below to the nearest fourth of a unit. [tex]x+5=-3^x+4[/tex] A. [tex]x \approx 3.75[/tex] B. [tex]x=-1.25[/tex] C. [tex]x=-2.25[/tex] D. [tex]x=1.25[/tex]
Answer (2)
Rewrite the equation as x + 1 + 3 x = 0 .
Define the function f ( x ) = x + 1 + 3 x .
Evaluate f ( x ) for x = 3.75 , − 1.25 , − 2.25 , 1.25 .
Find the value of x that makes f ( x ) closest to 0, which is x = − 1.25 . Therefore, the answer is x = − 1.25 .
Explanation
Understanding the Problem We are given the equation x + 5 = − 3 x + 4 and asked to find the solution to the nearest fourth of a unit using a table of values. The possible solutions are A. x ≈ 3.75 , B. x = − 1.25 , C. x = − 2.25 , D. x = 1.25 .
Rewriting the Equation First, let's rewrite the equation as x + 5 + 3 x − 4 = 0 , which simplifies to x + 1 + 3 x = 0 . We want to find the value of x that makes this expression equal to zero.
Defining the Function Define the function f ( x ) = x + 1 + 3 x . We will evaluate f ( x ) for the given values x = 3.75 , − 1.25 , − 2.25 , 1.25 to see which one is closest to 0.
Evaluating the Function Now, let's evaluate the function at each of the given points:
For x = 3.75 :
f ( 3.75 ) = 3.75 + 1 + 3 3.75 ≈ 3.75 + 1 + 50.3867 = 55.1367
For x = − 1.25 :
f ( − 1.25 ) = − 1.25 + 1 + 3 − 1.25 ≈ − 1.25 + 1 + 0.2533 = 0.0033
For x = − 2.25 :
f ( − 2.25 ) = − 2.25 + 1 + 3 − 2.25 ≈ − 2.25 + 1 + 0.0844 = − 1.1656
For x = 1.25 :
f ( 1.25 ) = 1.25 + 1 + 3 1.25 ≈ 1.25 + 1 + 4.1982 = 6.4482
Finding the Closest Solution We are looking for the value of x that makes f ( x ) closest to 0. Comparing the absolute values of the results:
∣ f ( 3.75 ) ∣ = 55.1367 ∣ f ( − 1.25 ) ∣ = 0.0033 ∣ f ( − 2.25 ) ∣ = 1.1656 ∣ f ( 1.25 ) ∣ = 6.4482
Since ∣ f ( − 1.25 ) ∣ = 0.0033 is the smallest, x = − 1.25 is the closest solution.
Final Answer Therefore, the correct answer is B. x = − 1.25 .
Examples
Imagine you're designing a seesaw where one side's height is linearly related to the position, and the other side's height changes exponentially. Finding the balance point is like solving the equation x + 5 = − 3 x + 4 . This type of problem appears in various scenarios, such as balancing chemical reactions, modeling population growth against resource depletion, or even designing financial models where investments grow exponentially while expenses increase linearly. Understanding how to solve such equations helps in making informed decisions and predictions in these contexts.
The solution to the equation x + 5 = − 3 x + 4 is found by evaluating the function f ( x ) = x + 1 + 3 x for given values. After evaluating, it was determined that the value x = − 1.25 brings f ( x ) closest to zero. Thus, the answer is B. x = − 1.25 .
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