\( \frac{3 p}{p-1}-\frac{2}{p-4}=\frac{2 p^{2}-11 p}{p^{2}-5 p+4} \) The solution set is

To solve the equation \( \frac{3 p}{p-1}-\frac{2}{p-4}=\frac{2 p^{2}-11 p}{p^{2}-5 p+4} \), we first notice that the right side can be simplified further. The denominator \( p^{2}-5p+4 \) can be factored into \( (p-1)(p-4) \). Once we rewrite the equation as: \[ \frac{3p}{p-1} - \frac{2}{p-4} = \frac{2p^2 - 11p}{(p-1)(p-4)} \] it's easier to manipulate. We’ll find a common denominator for the left side, which is also \( (p-1)(p-4) \): \[ \frac{3p(p-4) - 2(p-1)}{(p-1)(p-4)} = \frac{2p^2 - 11p}{(p-1)(p-4)} \] Now we can cross-multiply since the denominators are the same: \[ 3p(p-4) - 2(p-1) = 2p^2 - 11p \] Expanding both sides gives: \[ 3p^2 - 12p - 2p + 2 = 2p^2 - 11p \] \[ 3p^2 - 14p + 2 = 2p^2 - 11p \] Now, we can move all terms to one side: \[ 3p^2 - 2p^2 - 14p + 11p + 2 = 0 \] This simplifies to: \[ p^2 - 3p + 2 = 0 \] Factoring gives us: \[ (p-1)(p-2) = 0 \] Thus, the solutions are: \[ p = 1 \quad \text{or} \quad p = 2 \] Checking these solutions against the original equation ensures they do not create any undefined expressions due to the denominators. The solution set is \( \{1, 2\} \).