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Joel and Kevin are each putting money in a savings account to buy a new bicycle. The amount, in dollars, in Joel's savings account, [tex]$x$[/tex] weeks after the start of the year, is modeled by function [tex]$j$[/tex]. The amount of money in Kevin's account, at the same time, is modeled by function [tex]$k$[/tex].
[tex]\[
\begin{array}{l}
j(x) = 25 + 3x \\
k(x) = 15 + 2x
\end{array}
\][/tex]

Which function correctly represents how much more money, in dollars, is in Joel's account than in Kevin's account [tex]$x$[/tex] weeks after the start of the year?

A. [tex]\((j-k)(x) = 10 + 5x\)[/tex]

B. [tex]\((j-k)(x) = 10 + x\)[/tex]

C. [tex]\((j-k)(x) = 40 + 5x\)[/tex]

D. [tex]\((j-k)(x) = 40 + x\)[/tex]



Answer :

GinnyAnswer
To find out how much more money is in Joel's account compared to Kevin's account [tex]\(x\)[/tex] weeks after the start of the year, we need to calculate the difference between the two functions [tex]\( j(x) \)[/tex] and [tex]\( k(x) \)[/tex].

Given:
[tex]\[ j(x) = 25 + 3x \][/tex]
[tex]\[ k(x) = 15 + 2x \][/tex]

We need to find the function [tex]\((j-k)(x)\)[/tex], which represents the difference between [tex]\(j(x)\)[/tex] and [tex]\(k(x)\)[/tex].

[tex]\[ (j - k)(x) = j(x) - k(x) \][/tex]
[tex]\[ = (25 + 3x) - (15 + 2x) \][/tex]

Now, we will simplify the expression step-by-step:
1. Distribute the negative sign through the [tex]\(k(x)\)[/tex] term:
[tex]\[ = 25 + 3x - 15 - 2x \][/tex]

2. Combine like terms:
[tex]\[ = (25 - 15) + (3x - 2x) \][/tex]
[tex]\[ = 10 + x \][/tex]

So, the function that correctly represents how much more money is in Joel's account than in Kevin's account [tex]\(x\)[/tex] weeks after the start of the year is:
[tex]\[ (j - k)(x) = 10 + x \][/tex]

Therefore, the correct answer is:
B. [tex]\((j - k)(x) = 10 + x\)[/tex]

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