Solve the rational equation by multiplying both sides by the LCD.

Answer:

The rational equation is solved and the solutions are 7 and the extraneous solution is x = -2

What is a rational function equation?
A rational equation can be defined as an equation that involves at least one rational expression

A rational function is a function of the form f( x )=P( x ) / Q( x ) ,

f( x ) = P( x ) / Q( x ) , where P( x ) and Q( x ) are both polynomials.

A rational function f ( x )=P( x ) / Q( x )

f( x ) = P( x ) / Q( x ) may have a vertical asymptote

The domain is the set of all real numbers for which Q( x ) ≠ 0

Given data,

Let the rational equation be represented as A

Now, the value of A is

( 3 / ( x²+5x+6 ) ) + ( ( x-1) / (x+2) ) = 7 / (x+3) by equation (1)

On simplifying, we get

A rational function is a function of the form f( x )=P( x ) / Q( x ) ,

f( x ) = P( x ) / Q( x ) , where P( x ) and Q( x ) are both polynomials.

So, multiplying both sides by ( x²+5x+6 ), we get

3 + x² + 2x – 3 = 7x + 14 by equation (2)

On further simplification, we get

x² – 5x – 14 = 0

On factorizing the equation , we get

x² – 7x + 2x – 14 = 0

Taking the common factors, we get

x ( x – 7 ) + 2 ( x – 7 ) = 0

( x + 2 ) ( x – 7 ) = 0

So , the two solutions of x are x = 7 and x = -2

Hence , the solutions of rational equation are x = 7 and x = -2

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