Solve for x 5/(x+4)=4+3/(x-2)

Problem

5/(x+4)=4+3/(x−2)

Solution

1. Identify the common denominator for all terms in the equation, which is (x+4)*(x−2)

2. Multiply every term by the common denominator to eliminate the fractions.

(x+4)*(x−2)⋅5/(x+4)=(x+4)*(x−2)⋅4+(x+4)*(x−2)⋅3/(x−2)

3. Simplify the equation by canceling the denominators.

5*(x−2)=4*(x+4)*(x−2)+3*(x+4)

4. Expand the expressions on both sides of the equation.

5*x−10=4*(x2+2*x−8)+3*x+12

5*x−10=4*x2+8*x−32+3*x+12

5. Combine like terms on the right side.

5*x−10=4*x2+11*x−20

6. Rearrange into standard quadratic form a*x2+b*x+c=0 by moving all terms to one side.

0=4*x2+6*x−10

7. Divide by the common factor of 2 to simplify the quadratic.

0=2*x2+3*x−5

8. Factor the quadratic equation by finding two numbers that multiply to −10 and add to 3

0=(2*x+5)*(x−1)

9. Apply the zero product property to solve for x

2*x+5=0⇒x=−5/2

x−1=0⇒x=1

10. Check for extraneous solutions by ensuring the values do not make the original denominators zero. Since x≠−4 and x≠2 both solutions are valid.

Final Answer

5/(x+4)=4+3/(x−2)⇒x=−5/2,x=1

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