Question

(continued)

1. \\(\frac{x + 6}{x - 4} - \frac{30}{x^2 - 5x + 4} = \frac{3}{x - 1}\\)
2. \\(\frac{x - 2}{x - 4} - \frac{2}{x - 1} = \frac{12}{x^2 - 5x + 4}\\)

Explanation:

Step1: Factor the denominator

Factor \(x^2 - 5x + 4\) as \((x - 1)(x - 4)\). The equation becomes \(\frac{x + 6}{x - 4} - \frac{30}{(x - 1)(x - 4)} = \frac{3}{x - 1}\) (taking problem 10 as an example, same method for 12).

Step2: Find the LCD

The least common denominator (LCD) of \((x - 4)\), \((x - 1)(x - 4)\), and \((x - 1)\) is \((x - 1)(x - 4)\).

Step3: Multiply each term by LCD

Multiply each term by \((x - 1)(x - 4)\): \((x + 6)(x - 1) - 30 = 3(x - 4)\).

Step4: Expand and simplify

Expand \((x + 6)(x - 1)\) to \(x^2 + 5x - 6\). So the equation is \(x^2 + 5x - 6 - 30 = 3x - 12\), which simplifies to \(x^2 + 5x - 36 = 3x - 12\), then \(x^2 + 2x - 24 = 0\).

Step5: Solve the quadratic equation

Factor \(x^2 + 2x - 24\) as \((x + 6)(x - 4) = 0\). So \(x = -6\) or \(x = 4\). But \(x = 4\) makes the original denominator zero, so we discard it. Thus, the solution for problem 10 is \(x = -6\).

For problem 12, following similar steps:

Step1: Factor the denominator

\(x^2 - 5x + 4=(x - 1)(x - 4)\). The equation is \(\frac{x - 2}{x - 4} - \frac{2}{x - 1} = \frac{12}{(x - 1)(x - 4)}\).

Step2: Multiply by LCD \((x - 1)(x - 4)\)

\((x - 2)(x - 1) - 2(x - 4) = 12\).

Step3: Expand and simplify

Expand \((x - 2)(x - 1)\) to \(x^2 - 3x + 2\). The equation becomes \(x^2 - 3x + 2 - 2x + 8 = 12\), then \(x^2 - 5x - 2 = 0\)? Wait, no, let's recalculate: \(x^2 - 3x + 2 - 2x + 8 = 12\) → \(x^2 - 5x + 10 = 12\) → \(x^2 - 5x - 2 = 0\)? Wait, no, maybe I made a mistake. Wait, \(-2(x - 4)= -2x + 8\), so \(x^2 - 3x + 2 - 2x + 8 = 12\) → \(x^2 - 5x + 10 - 12 = 0\) → \(x^2 - 5x - 2 = 0\)? Wait, no, let's do it again. \((x - 2)(x - 1)=x^2 - 3x + 2\), subtract \(2(x - 4)=2x - 8\) (wait, no, it's \(-2(x - 4)= -2x + 8\)), so \(x^2 - 3x + 2 - 2x + 8 = 12\) → \(x^2 - 5x + 10 = 12\) → \(x^2 - 5x - 2 = 0\). Using quadratic formula \(x=\frac{5\pm\sqrt{25 + 8}}{2}=\frac{5\pm\sqrt{33}}{2}\)? Wait, maybe I messed up the sign. Wait, original equation: \(\frac{x - 2}{x - 4} - \frac{2}{x - 1} = \frac{12}{(x - 1)(x - 4)}\). Multiply by \((x - 1)(x - 4)\): \((x - 2)(x - 1) - 2(x - 4) = 12\). \((x^2 - 3x + 2) - 2x + 8 = 12\) → \(x^2 - 5x + 10 = 12\) → \(x^2 - 5x - 2 = 0\). The discriminant is \(25 + 8 = 33\), so \(x=\frac{5\pm\sqrt{33}}{2}\). Wait, but maybe I made a mistake in the problem 12 setup. Alternatively, maybe the user wants solution for one of them. Let's confirm problem 10 solution again. For problem 10, after getting \(x = -6\) and \(x = 4\), \(x = 4\) is extraneous, so \(x = -6\) is valid.

Answer:

(for problem 10):
\(x = -6\)

(For problem 12, the solution is \(x=\frac{5\pm\sqrt{33}}{2}\) if we follow the steps, but maybe there was a miscalculation. Alternatively, if there's a typo, but based on the given problem, the above is the solution process.)