Given the functions [tex]f(x)=\frac{1}{x+2}[/tex] and [tex]g(x)=\frac{3}{2x-1}[/tex], find the value of [tex]x[/tex] such that [tex]f(x)=g(x)[/tex].



Answer :

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]