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Determining a Linear Equation to Find an Input Value

Consider [tex][tex]$f(x)=1.8 x-10$[/tex][/tex] and [tex][tex]$g(x)=-4$[/tex][/tex].

[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
-4 & -17.2 \\
\hline
-2 & -13.6 \\
\hline
0 & -10 \\
\hline
2 & -6.4 \\
\hline
4 & -2.8 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$g(x)$[/tex] \\
\hline
-4 & -4 \\
\hline
-2 & -4 \\
\hline
0 & -4 \\
\hline
2 & -4 \\
\hline
4 & -4 \\
\hline
\end{tabular}
\][/tex]

Select the equation that can be used to find the input value at which [tex][tex]$f(x)=g(x)$[/tex][/tex], and then use that equation to find the input, or [tex][tex]$x$[/tex][/tex]-value.

A. [tex][tex]$1.8 x - 10 = -4 ; x = \frac{10}{3}$[/tex][/tex]

B. [tex][tex]$1.8 x = -4 ; x = -\frac{20}{9}$[/tex][/tex]

C. [tex][tex]$1.8 x - 10 = -4 ; x = -\frac{10}{3}$[/tex][/tex]

D. [tex][tex]$-4 = x$[/tex][/tex]



Answer :

GinnyAnswer
To determine the input value [tex]\( x \)[/tex] at which [tex]\( f(x) \)[/tex] equals [tex]\( g(x) \)[/tex], we need to solve the equation [tex]\( f(x) = g(x) \)[/tex].

Given the functions:
[tex]\[ f(x)=1.8x-10 \][/tex]
[tex]\[ g(x)=-4 \][/tex]

To find the input value where these two functions are equal, set [tex]\( f(x) \)[/tex] equal to [tex]\( g(x) \)[/tex]:

[tex]\[ f(x) = g(x) \][/tex]
[tex]\[ 1.8x - 10 = -4 \][/tex]

Now, solve this equation step-by-step:

1. Add 10 to both sides to isolate the term with [tex]\( x \)[/tex] on one side:
[tex]\[ 1.8x - 10 + 10 = -4 + 10 \][/tex]
[tex]\[ 1.8x = 6 \][/tex]

2. Divide both sides by 1.8 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{6}{1.8} \][/tex]

Simplify the fraction:
[tex]\[ x = \frac{6}{1.8} = \frac{60}{18} = \frac{10}{3} \][/tex]

Therefore, the input value [tex]\( x \)[/tex] at which [tex]\( f(x) = g(x) \)[/tex] is:
[tex]\[ x = \frac{10}{3} \][/tex]

Thus, the correct answer is:
[tex]\[ 1.8x - 10 = -4; x = \frac{10}{3} \][/tex]

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