Question

Solve the following rational equation. Be sure to list the NPV’s and verify your solution(s).


`(2)/(2x-6)-(x)/(x^2-9) =(3)/(x-3)`

Answer 1

• NPV: ±3
• x = -2

Explanation:

You want the solutions to the rational equation ...

`(2)/(2x-6)-(x)/(x^2-9)=(3)/(x-3)`

Rewrite

We choose to rewrite the equation in the form f(x) = 0. We can start by subtracting the term on the right side (from both sides).

`(2)/(2x-6)-(x)/(x^2-9)-(3)/(x-3)=0\\\\(2)/(2(x-3))-(x)/((x-3)(x+3))-(3)/(x-3)=0\\\\\\((x+3)-x-3(x+3))/((x-3)(x+3))=0\qquad\text{cancel 2 in first term; common denominator}\\\\\\(-3x-6)/((x-3)(x+3))=0\qquad\text{collect terms}`

Solution

The "not-possible values" will be values of x that make the denominator zero: x = ±3. The solution is ...

`-3(x+2)=0\qquad\text{numerator must be zero}\\\\\boxed{x=-2}\qquad\text{divide by -3, subtract 2}`

Check

In the original equation, we have ...

`(2)/(2(-2)-6)-(-2)/((-2)^2-9)=(3)/((-2)-3)\\\\(2)/(-10)-(-2)/(-5)=(3)/(-5)\\\\-(1)/(5)-(2)/(5)=-(3)/(5)\qquad\text{TRUE}`

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Additional comment

We like to write the equation as a single fraction compared to zero so that any common factors can be cancelled from numerator and denominator. This helps prevent extraneous solutions. Here, there are none.