- Solve the first inequality \(2p - \frac{p-2}{5} > 4\) to find \(p > 2\).
- Solve the second inequality \(\frac{p}{2} - \frac{p}{8} \leq 6\) to find \(p \leq 16\).
- Find the intersection of both solution sets to satisfy the system.
- The combined solution for the system is \(\boxed{2 < p \leq 16}\).
Explanation
- Analysis of the System of Inequalities
The problem requires finding the solution set for the variable \(p\) that satisfies the following system of linear inequalities:
\[ \left\{\begin{array}{l} 2p - \frac{p-2}{5} > 4 \ \frac{p}{2} - \frac{p}{8} \leq 6 \end{array}\right. \]
We will solve each inequality individually to determine the range of values for \(p\) in each case, and then find the intersection of these two ranges to obtain the final solution set.
- Solving the First Inequality
We start by solving the first inequality:
\[2p - \frac{p-2}{5} > 4\]
To eliminate the fraction, we multiply every term by 5:
\[5 \times (2p) - (p - 2) > 5 \times 4\]
\[10p - p + 2 > 20\]
Next, simplify the terms:
\[9p + 2 > 20\]
Subtract 2 from both sides:
\[9p > 18\]
Divide by 9:
\[p > 2\]
So, the first inequality is satisfied for all \(p > 2\).
- Solving the Second Inequality
Now we solve the second inequality:
\[\frac{p}{2} - \frac{p}{8} \leq 6\]
To subtract these fractions, we find a common denominator, which is 8:
\[\frac{4p}{8} - \frac{p}{8} \leq 6\]
\[\frac{3p}{8} \leq 6\]
Multiply both sides by 8 to clear the denominator:
\[3p \leq 48\]
Divide by 3:
\[p \leq 16\]
So, the second inequality is satisfied for all \(p \leq 16\).
- Combining Results and Final Solution
Now we combine the results from the two inequalities. The system is satisfied if and only if both conditions are met:
- \(p > 2\)
- \(p \leq 16\)
Combining these two requirements gives us the interval:
\[2 < p \leq 16\]
This interval represents all values of \(p\) that satisfy the original system. Thus, the solution set is \(\boxed{2 < p \leq 16}\).
Examples
In engineering, specific parameters must stay within safety ranges. For example, a machine component might require a pressure \(p\) (in bars) to be greater than 2 to function properly (to avoid stalling), but it must also be less than or equal to 16 to prevent damage from excessive force. By solving a system of inequalities, engineers can determine the "operating window" where the machine runs safely and effectively. This mathematical approach helps maintain equipment integrity and operational efficiency in various real-world industrial systems.