Okay, so I need to find all values of p that satisfy both inequalities: 0 ≥ 54p - 144 and 0 > 12 - 20p. Hmm, let me start by tackling each inequality separately and then find the intersection of the solutions. 

First, let's look at the first inequality: 0 ≥ 54p - 144. I should solve for p here. To do that, I can add 144 to both sides to isolate the term with p. That would give me 144 ≥ 54p. Then, divide both sides by 54 to get p by itself. Let me compute that. 144 divided by 54... Well, 54 times 2 is 108, and 54 times 2.666... is 144. So 144/54 simplifies. Let me divide numerator and denominator by 18: 144 ÷ 18 = 8, 54 ÷ 18 = 3. So that's 8/3. Wait, 8/3 is approximately 2.666..., right? So p ≤ 8/3. But wait, when I divide both sides by 54, which is positive, the inequality sign remains the same. So yes, p ≤ 8/3.

Now, the second inequality: 0 > 12 - 20p. Let me solve this one. First, subtract 12 from both sides. That gives -12 > -20p. Then, divide both sides by -20. But wait, when you divide or multiply both sides of an inequality by a negative number, you have to reverse the inequality sign. So dividing by -20, the inequality becomes -12 / -20 < p. Simplifying that, -12 divided by -20 is 12/20, which reduces to 3/5. So p > 3/5.

So now I have two inequalities: p ≤ 8/3 and p > 3/5. The solution set is the intersection of these two, which is 3/5 < p ≤ 8/3. To express this in interval notation, it would be (3/5, 8/3]. 

Wait, let me double-check my steps to make sure I didn't make a mistake. For the first inequality: 0 ≥ 54p - 144. Adding 144 to both sides: 144 ≥ 54p. Dividing by 54: 144/54 = 8/3. So p ≤ 8/3. Correct. For the second inequality: 0 > 12 - 20p. Subtract 12: -12 > -20p. Divide by -20 and reverse inequality: 12/20 < p, which is 3/5 < p. So p > 3/5. So yes, the intersection is 3/5 < p ≤ 8/3. 

Expressing in interval notation, since 3/5 is approximately 0.6 and 8/3 is approximately 2.666..., the interval is open at 3/5 and closed at 8/3. So (3/5, 8/3]. 

I think that's all. Let me check with a test value. Let's pick p = 1, which is between 0.6 and 2.666. For the first inequality: 54*1 -144 = 54 -144 = -90. 0 ≥ -90, which is true. For the second inequality: 12 -20*1 = 12 -20 = -8. 0 > -8, which is also true. So p=1 is valid. 

Another test: p = 0.5, which is below 3/5. First inequality: 54*0.5 -144 = 27 -144 = -117. 0 ≥ -117, true. Second inequality: 12 -20*0.5 = 12 -10 = 2. 0 > 2 is false. So p=0.5 is excluded, which is correct. 

Another test: p = 8/3 ≈ 2.666. First inequality: 54*(8/3) -144 = 54*(8/3) = 18*8 = 144. 144 -144 = 0. 0 ≥ 0 is true. Second inequality: 12 -20*(8/3) = 12 - 160/3 = (36 -160)/3 = (-124)/3 ≈ -41.333. 0 > -41.333 is true. So p=8/3 is included. 

And a point above 8/3, say p=3. First inequality: 54*3 -144 = 162 -144 = 18. 0 ≥ 18 is false. So p=3 is excluded, which is correct. 

Therefore, the solution set is indeed all real numbers p such that 3/5 < p ≤ 8/3. In interval notation, that's (3/5, 8/3]. 

I think that's thorough. I considered solving each inequality step by step, checked the critical points, tested values within and outside the interval to ensure correctness. No mistakes detected in the process. So the answer should be as stated.
Thus, the solution in interval notation is $\boxed{\left( \frac{3}{5}, \frac{8}{3} \right]}$.
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