To solve the inequality, we will first find a common denominator for all terms and then clear the fractions. The given inequality is: $$ -1 + \frac{2}{3p} \le \frac{1}{4} - \frac{5}{6p} $$ We are given that $p > 0$. Step 1: Find the least common multiple (LCM) of the denominators $1, 3p, 4,$ and $6p$. The LCM of $1, 3p, 4, 6p$ is $12p$. Step 2: Multiply every term in the inequality by the LCM, $12p$. Since $p > 0$, $12p$ is positive, so the inequality sign will not change direction. $$ 12p(-1) + 12p\left(\frac{2}{3p}\right) \le 12p\left(\frac{1}{4}\right) - 12p\left(\frac{5}{6p}\right) $$ Step 3: Simplify each term. $$ -12p + \frac{24p}{3p} \le \frac{12p}{4} - \frac{60p}{6p} $$ $$ -12p + 8 \le 3p - 10 $$ Step 4: Gather terms with $p$ on one side and constant terms on the other side. Add $12p$ to both sides: $$ 8 \le 3p + 12p - 10 $$ $$ 8 \le 15p - 10 $$ Add $10$ to both sides: $$ 8 + 10 \le 15p $$ $$ 18 \le 15p $$ Step 5: Solve for $p$. Divide both sides by $15$: $$ \frac{18}{15} \le p $$ Step 6: Simplify the fraction. $$ \frac{6}{5} \le p $$ This can also be written as $p \ge \frac{6}{5}$. Since $\frac{6}{5} = 1.2$, and $1.2 > 0$, the condition $p > 0$ is satisfied. The solution for $p$ is: $$ \boxed{p \ge \frac{6}{5}} $$