Solving for ( g(x) ): Step-by-Step Guide to Advanced Algebraic Manipulations

Algebraic manipulation involves solving complex equations and expressions to find the value of a function. In this guide, we will solve for ( g(x) ) given the equations ( fleft(frac{x}{x-2}right) 2left(frac{x-2}{x^5}right) ) and ( f(x) frac{3}{x-2} ). This process will involve several key steps, including substitution and algebraic transformations.

Step 1: Substitution and Simplification

Let ( t frac{x}{x - 2} ).

Solve for ( x ):

( x cdot t - 2t x )

( xt - x 2t )

( xt - 1 2t )

( x frac{2t}{t - 1} )

Step 2: Substitute and Simplify Further

Substitute this expression into the first given equation:

( fleft(frac{frac{2t}{t - 1}}{frac{2t}{t - 1} - 2}right) frac{2left(frac{2t}{t - 1}right) - 4}{frac{2t}{t - 1} - 5} )

( t frac{2t}{2t - 2t 2} )

( t frac{22t - 4t - 1}{2t - 5t 5} )

( t frac{4}{7t - 5} )

Step 3: Solve the Equation for ( g(x) )

Next, we solve for ( g(t) ) using the definition of ( f(x) ) and equate it to the last result:

( f(t) frac{3}{t - 2} frac{4}{7t - 5} )

( 3(7t - 5) 4(t - 2) )

( 21t - 15 4t - 8 )

( 21t - 4t -8 15 )

( 17t 7 )

( t frac{7}{17} )

Substitute back to find ( g(x) ):

( g(x) frac{21x - 7}{4} )

Conclusion

By following these meticulous steps, we have successfully derived the expression for ( g(x) ). The solved expression is:

( g(x) frac{21x - 7}{4} )

If you verify this solution and follow the correct steps, the result will indeed hold true.

Additional Tips and Insights

Algebraic manipulation is a powerful tool in solving complex equations. Practice this method with different expressions to build fluency and intuition.

Knowing the key steps in solving such equations can help you tackle more advanced algebraic problems in the future.