Sal R.

asked • 06/12/16# rational equation questions

what is the solution for to the rational equation

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

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## 1 Expert Answer

Darryl K. answered • 06/12/16

Experienced Math Tutor

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)/(x

^{2}+10x+24)Step 1) Factor all denominators and find the restrictions on x. (x+4) and (x+6) are already factored. x

^{2}+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.

x

^{2}+ 4x -12 +x^{2}+5x + 4 = 11x + 322x

^{2}+ 13x - 8 = 11x + 32 Moving all terms to the left2x

^{2}+2x - 40 = 0 Dividing each term by 2x

^{2}+ x - 20 = 0Since 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.

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Kenneth S.

06/12/16