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# Solve the following rational equation in five steps. Show all your work:

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Solve the following rational equation in five steps. Show all your work:

x+6/x-3=2x/x-3

1.Multiply the equation by the LCD to eliminate fractions. Show your work and state the resulting quadratic equation in standard form.

2.Solve this Quadratic by factoring showing all your work. State the solutions in simplified form.

3.Are any of these extraneous solutions? Check your work and state the solution set of the original equation, if any.

Dec 20, 2017

#1
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x+6/x-3=2x/x-3   Cross multiply

(x + 6) (x - 3) = 2 x (x - 3)

x (x + 3) - 18 =2x(x - 3)

x^2 + 3x -18 =2x^2 - 6x

x^2 + 3x - 18 - 2x^2 + 6x =0

-x^2 + 9x - 18 = 0, solve

Factor the left-hand side.
The left-hand side factors into a product with three terms:
-((x - 6) (x - 3)) = 0

Multiply both sides by a constant to simplify the equation.
Multiply both sides by -1:
(x - 6) (x - 3) = 0

Find the roots of each term in the product separately.
Split into two equations:
x - 6 = 0 or x - 3 = 0

Look at the first equation: Solve for x.
x = 6             or             x = 3

Dec 20, 2017
edited by Guest  Dec 20, 2017
edited by Guest  Dec 20, 2017
#2
+101750
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Thanks, Guest.....a slight correction....only x  = 6  is a solution...x  = 3  makes an original denominator 0, so that is not allowed

Dec 20, 2017
#3
+2340
+2

$$\frac{x+6}{x-3}=\frac{2x}{x-3}$$

Since the denominators are already the same, all you need to do is set the numerators equal to each other.

$$x+6=2x\\ 6=x$$

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Dec 20, 2017