Find the value of $t$ that satisfies $\frac{1}{t+2} + \frac{2t}{t+2} - \frac{3}{t+2} = 3$.
I notice that all the fractions in the equation have the same denominator, $t+2$. This makes it easier to simplify the equation by multiplying both sides by $t+2$ and cancelling out the denominators. Doing that, I get $1 + 2t - 3 = 3(t+2)$. Simplifying further, I get $2t - 2 = 3t + 6$. Subtracting $2t$ from both sides, I get $-2 = t + 6$. Subtracting $6$ from both sides, I get $t = -8$. This is the value of $t$ that satisfies the equation. 