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Betty has a checking account and a non-interest bearing savings account. The function [tex]$C(x)$[/tex], shown below, represents her checking account balance, where [tex]$x$[/tex] is the number of months. The function [tex]$S(x)$[/tex], shown below, represents her saving account balance.

[tex]\[
\begin{array}{l}
C(x) = -|240x - 140| + 1,400 \\
S(x) = 450x + 100
\end{array}
\][/tex]

Which function represents Betty's total account balance, [tex]\( T(x) \)[/tex]?

A. [tex]\(\pi(x) = |240x + 140| + 450x + 1,500\)[/tex]

B. [tex]\(\pi(x) = -|240x - 140| + 450x + 1,300\)[/tex]

C. [tex]\(T(x) = |240x - 140| + 450x + 1,500\)[/tex]

D. [tex]\(T(x) = -|240x - 140| + 450x + 1,400\)[/tex]



Answer :

GinnyAnswer
Let's start by understanding the two given functions, [tex]\( C(x) \)[/tex] and [tex]\( S(x) \)[/tex]:

1. The function [tex]\( C(x) \)[/tex] represents Betty's checking account balance:
[tex]\[ C(x) = -|240x - 140| + 1400 \][/tex]

2. The function [tex]\( S(x) \)[/tex] represents Betty's savings account balance:
[tex]\[ S(x) = 450x + 100 \][/tex]

To find Betty's total account balance, [tex]\( T(x) \)[/tex], we need to add these two functions together:

[tex]\[ T(x) = C(x) + S(x) \][/tex]

Substituting the given functions into this equation, we get:

[tex]\[ T(x) = \left( -|240x - 140| + 1400 \right) + \left( 450x + 100 \right) \][/tex]

Combining like terms, we obtain:

[tex]\[ T(x) = -|240x - 140| + 450x + 1400 + 100 \][/tex]

Simplifying further, we get:

[tex]\[ T(x) = -|240x - 140| + 450x + 1500 \][/tex]

With this form, we can match it against the provided options:

1. [tex]\( \pi(x) = |240x + 140| + 450x + 1500 \)[/tex]
2. [tex]\( \pi(x) = -|240x - 140| + 450x + 1300 \)[/tex]
3. [tex]\( T(x) = |240x - 140| + 450x + 1500 \)[/tex]
4. [tex]\( T(x) = -|240x - 140| + 450x + 1400 \)[/tex]

By comparing these, we see that the correct function representing Betty's total account balance is:

[tex]\[ T(x) = -|240x - 140| + 450x + 1500 \][/tex]

Hence, the correct option is:
[tex]\[ \boxed{T(x) = -|240x - 140| + 450x + 1500} \][/tex]