rational equation questions

Question

what is the solution for to the rational equation

x-2/x+4+x+1/x+6=11x+32/x^2+10x+24

 

 

Answer 1

I have done many problems of this type and tutor students who leave parenthesis off when doing test.  Unfortunately if you do it on a test it results in wrong answers and a low grade.  To me the problem is

 

(x-2)/(x+4) + (x+1)/(x+6) = (11x+32)/(x²+10x+24)

 

Step 1) Factor all denominators and find the restrictions on x.  (x+4) and (x+6) are already factored.  x²+10x+24 = (x+4)(x+6)

 

(x-2)/(x+4) + (x+1)/(x+6) = (11x+32)/((x+4)(x+6))

 

We know we cannot divide by zero.  We see from above that x cannot equal -4 and -6.

 

Step 2) Find the LCD.  We see that the LCD = (x+4)(x+6).

 

Step 3)  Multiply each term by the LCD.  This will clear all denominators.

 

(x-2)(x+4)(x+6)/(x+4) + (x+1)(x+4)(x+6)/(x+6) = (11x+32)(x+4)(x+6)/((x+4)(x+6))

 

Step 4) Cancel.

 

(x-2)(x+6) + (x+1)(x+4) = 11x + 32

 

Step 5)  Solve for x by expanding and simplifying.

 

x² + 4x -12 +x² +5x + 4 = 11x + 32

 

2x² + 13x - 8 = 11x + 32     Moving all terms to the left

 

2x² +2x - 40 = 0    Dividing each term by 2

 

x² + x - 20 = 0

 

Since this is a quadratic equation we can either factor or use the quadratic formula.

 

Factoring gives

 

(x + 5)(x - 4) = 0

 

x + 5 = 0 → x = -5     or     x - 4 = 0 → x = 4

 

Step 6) Check your solutions to make sure that none are extraneous.

 

Since x does not equal -4 or -6 we see that both solutions are valid.