<Question>
Find all $p$ which satisfy both the inequalities $0\ge 54p-144$ and $0>12-20p$. Express your answer in interval notation, reducing any fractions in your answer.
</Question>

<think>
<Plan_1>
Okay, so I need to find all the values of p that satisfy both inequalities: 0 ≥ 54p - 144 and 0 > 12 - 20p. Hmm, let me start by solving each inequality separately first. Maybe once I have the solutions for each, I can find where they overlap, right? That should give me the final answer. Let me take it step by step.

Starting with the first inequality: 0 ≥ 54p - 144. I want to solve for p here. So, let me rewrite this inequality to make it a bit clearer.
If I add 144 to both sides, that should isolate the term with p. Let's see:

<prompt_1.1> Solve the first inequality \( 0 \ge 54p - 144 \) for \( p \) as per the above plan. </prompt_1.1>

Now, moving on to the second inequality: 0 > 12 - 20p. Again, I need to solve for p. Let me rearrange this. Maybe subtract 12 from both sides first?
Wait, but if I do that, let's see:

<prompt_1.2> Solve the second inequality \( 0 > 12 - 20p \) for \( p \) as per the above plan. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
0 + 144 ≥ 54p - 144 + 144

Which simplifies to:

144 ≥ 54p

Now, to solve for p, I should divide both sides by 54. Let me do that:

144 / 54 ≥ p

Simplifying 144 divided by 54. Hmm, both numbers are divisible by 18. 144 ÷ 18 is 8, and 54 ÷ 18 is 3. So, 8/3. Therefore:

144 / 54 = 8/3 ≥ p

Which can also be written as:

p ≤ 8/3

Okay, so the first inequality tells me that p is less than or equal to 8/3.
</execution_1.1>

<execution_1.2>
So, 0 - 12 > -20p, which simplifies to -12 > -20p. Then, if I divide both sides by -20, I have to remember to reverse the inequality sign because I'm dividing by a negative number. So, dividing both sides by -20 gives me (-12)/(-20) < p. Simplifying that fraction, 12/20 reduces to 3/5, so that's 3/5 < p. Which is the same as p > 3/5.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.2:
Wait, let me check that again. If I have -12 > -20p, multiplying both sides by -1 (which flips the inequality) gives 12 < 20p. Then dividing both sides by 20: 12/20 < p.

<prompt_2.1> Verify the solution of the second inequality and correct the direction of the inequality if necessary. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Which simplifies to 3/5 < p. So, yes, p is greater than 3/5. So the second inequality gives p > 3/5.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_1.1, execution_2.1:
Now, combining both inequalities. The first one says p ≤ 8/3, and the second one says p > 3/5. So, the solution set is all p such that 3/5 < p ≤ 8/3.

But let me make sure I didn't make any mistakes in the algebra. Let's double-check the first inequality again.

<prompt_3.1> Recheck the solution of the first inequality. </prompt_3.1>

Second inequality: 0 > 12 - 20p. Let me plug in p = 3/5.

<prompt_3.2> Recheck the solution of the second inequality by plugging in values around the boundary. </prompt_3.2>

So, combining them, p has to be greater than 3/5 but less than or equal to 8/3. Now, converting 3/5 and 8/3 into decimals might help visualize the interval.
3/5 is 0.6, and 8/3 is approximately 2.666...., so the interval is from 0.6 to 2.666...., not including 0.6 but including 2.666...

But the problem says to express the answer in interval notation, reducing any fractions. So, I need to write it as (3/5, 8/3].

Let me just confirm if the endpoints are correctly included or excluded. For the first inequality, it's p ≤ 8/3, so 8/3 is included. For the second inequality, p > 3/5, so 3/5 is excluded.

<prompt_3.3> Confirm the interval notation for the solution set. </prompt_3.3>
</Plan_3>

<Execution_3>
<execution_3.1>
0 ≥ 54p - 144

Adding 144: 144 ≥ 54p

Divide by 54: 144/54 ≥ p. 144 divided by 54, as I did before, is 8/3. So yes, p ≤ 8/3.
</execution_3.1>

<execution_3.2>
12 - 20*(3/5) = 12 - 12 = 0. But the inequality is 0 > 12 - 20p, so when p is 3/5, the right side is 0, which is not greater than 0. Hence, p must be greater than 3/5.

If p is slightly more than 3/5, say 0.7, then 12 - 20*0.7 = 12 - 14 = -2, which is less than 0, so 0 > -2 is true. So that works. If p is exactly 3/5, we get 0, which doesn't satisfy the strict inequality.
</execution_3.2>

<execution_3.3>
Therefore, the interval is open at 3/5 and closed at 8/3. So, (3/5, 8/3].

Is there any chance I messed up the inequality signs? Let me check with a test value in each interval.

For example, take p = 1, which is in the interval (3/5, 8/3]. Let's check both inequalities.

First inequality: 54*1 - 144 = 54 - 144 = -90. Then 0 ≥ -90, which is true. Second inequality: 12 - 20*1 = 12 - 20 = -8. Then 0 > -8, which is true. So p = 1 works.

Now, take p = 8/3, which is approximately 2.666.... Let's check both inequalities.

First inequality: 54*(8/3) - 144 = 54*(8/3) = 18*8 = 144. So 144 - 144 = 0. Then 0 ≥ 0, which is true. Second inequality: 12 - 20*(8/3) = 12 - 160/3 = (36/3 - 160/3) = (-124/3). Then 0 > -124/3, which is true. So p = 8/3 is included.

Now, take a number greater than 8/3, say p = 3. Let's check the first inequality: 54*3 - 144 = 162 - 144 = 18. Then 0 ≥ 18? No, that's false. So p = 3 is excluded, which makes sense.

Now, take p = 3/5, which is 0.6. Let's check the second inequality: 12 - 20*(3/5) = 12 - 12 = 0. Then the inequality is 0 > 0, which is false. So p = 3/5 is excluded, which is correct.

Another test point between 3/5 and 8/3, say p = 1.5 (which is 3/2). First inequality: 54*(1.5) - 144 = 81 - 144 = -63. 0 ≥ -63, true. Second inequality: 12 - 20*(1.5) = 12 - 30 = -18. 0 > -18, true. So p = 1.5 works.

What about p = 0? That's less than 3/5. First inequality: 0 ≥ 54*0 -144 → 0 ≥ -144, true. Second inequality: 0 > 12 - 0 → 0 > 12, false. So p = 0 is excluded, which is correct.

And p = 8/3 is included, as we saw earlier. So the interval (3/5, 8/3] seems correct.
</execution_3.3>
</Execution_3>

<Plan_4>
Based on execution_3.3:
I think that's all. Let me just recap the steps:

<prompt_4.1> Recap the steps taken to solve the inequalities and confirm the solution. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
1. Solve each inequality separately.
2. Find the intersection of the two solution sets.
3. Express the final answer in interval notation with fractions reduced.

No steps seem missed here. All test points checked out, so confident the answer is correct.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the solution in interval notation is $\boxed{\left( \frac{3}{5}, \frac{8}{3} \right]}$.
</Final_answer>
</think>