To find the value of [tex]\( x \)[/tex] such that [tex]\( f(x) = g(x) \)[/tex], we need to set the functions equal to each other and solve for [tex]\( x \)[/tex].
Given:
[tex]\[ f(x) = \frac{1}{x + 2} \][/tex]
[tex]\[ g(x) = \frac{3}{2x - 1} \][/tex]
We need to solve the equation:
[tex]\[ \frac{1}{x + 2} = \frac{3}{2x - 1} \][/tex]
Step-by-Step Solution:
1. Cross-Multiply to clear the fractions:
[tex]\[
1 \cdot (2x - 1) = 3 \cdot (x + 2)
\][/tex]
2. Distribute the terms:
[tex]\[
2x - 1 = 3x + 6
\][/tex]
3. Move all terms involving [tex]\( x \)[/tex] to one side of the equation:
[tex]\[
2x - 3x = 6 + 1
\][/tex]
4. Combine like terms:
[tex]\[
-x = 7
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -7
\][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies [tex]\( f(x) = g(x) \)[/tex] is:
[tex]\[ x = -7 \][/tex]
