Question

question 2 of 10, step 1 of 1 solve the following rational equation and simplify your answer. \\(\frac{2x}{x + 2} = 2 - \frac{2}{x - 4}\\) answer how to enter your answer (opens in new window) separate multiple answers with commas. if the equation has no solution, indicate
o solution\. selecting a radio button will replace the entered answer value(s) with the radio button value. if the radio button is not selected, the entered answer is used. \\(x = \\) \\(\circ\\) no solution

Explanation:

Step1: Eliminate denominators

Multiply both sides by $(x+2)(x-4)$:
$$2x(x-4) = 2(x+2)(x-4) - 2(x+2)$$

Step2: Expand all terms

$$2x^2 - 8x = 2(x^2 - 2x - 8) - 2x - 4$$
$$2x^2 - 8x = 2x^2 - 4x - 16 - 2x - 4$$

Step3: Simplify right-hand side

$$2x^2 - 8x = 2x^2 - 6x - 20$$

Step4: Cancel $2x^2$ and solve for $x$

Subtract $2x^2$ from both sides:
$$-8x = -6x - 20$$
$$-8x + 6x = -20$$
$$-2x = -20$$
$$x = 10$$

Step5: Verify solution

Check $x=10$ does not make original denominators zero: $10+2=12
eq0$, $10-4=6
eq0$. Substitute into original equation:
Left: $\frac{2*10}{10+2}=\frac{20}{12}=\frac{5}{3}$
Right: $2-\frac{2}{10-4}=2-\frac{2}{6}=2-\frac{1}{3}=\frac{5}{3}$
Both sides are equal, so $x=10$ is valid.

Answer:

$x=10$