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]
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]
