If [tex]\( f(x) = 16x - 30 \)[/tex] and [tex]\( g(x) = 14x - 6 \)[/tex], for which value of [tex]\( x \)[/tex] does [tex]\( (f - g)(x) = 0 \)[/tex]?

A. [tex]\(-18\)[/tex]
B. [tex]\(-12\)[/tex]
C. 12
D. 18



Answer :

Certainly! Let's solve the problem step-by-step. We are given two functions [tex]\( f(x) = 16x - 30 \)[/tex] and [tex]\( g(x) = 14x - 6 \)[/tex]. We need to find the value of [tex]\( x \)[/tex] for which [tex]\( (f - g)(x) = 0 \)[/tex].

1. Define the combined function [tex]\( (f - g)(x) \)[/tex]:
\[tex]\[ (f - g)(x) = f(x) - g(x) \\][/tex]

2. Substitute the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] into the combined function:
\[tex]\[ (f - g)(x) = (16x - 30) - (14x - 6) \\][/tex]

3. Simplify the expression:
\[tex]\[ (f - g)(x) = 16x - 30 - 14x + 6 \\][/tex]
Combine like terms:
\[tex]\[ (f - g)(x) = (16x - 14x) + (-30 + 6) \\][/tex]
\[tex]\[ (f - g)(x) = 2x - 24 \\][/tex]

4. Set the combined function equal to 0:
\[tex]\[ 2x - 24 = 0 \\][/tex]

5. Solve for [tex]\( x \)[/tex]:
Add 24 to both sides of the equation:
\[tex]\[ 2x - 24 + 24 = 0 + 24 \\][/tex]
\[tex]\[ 2x = 24 \\][/tex]
Now, divide both sides by 2:
\[tex]\[ x = \frac{24}{2} \\][/tex]
\[tex]\[ x = 12 \\][/tex]

Therefore, the value of [tex]\( x \)[/tex] for which [tex]\( (f - g)(x) = 0 \)[/tex] is:
\[tex]\[ \boxed{12} \\][/tex]