- Start by simplifying the inequality \(2p - 8 < 5p + 4\) to isolate \(p\).
- Subtract \(2p\) and \(4\) from both sides to arrive at \(-12 < 3p\).
- Divide by \(3\) to find the simplified condition \(p > -4\).
- Test each value in the set \(S = \{-2, -3, -4, -5\}\) and observe that only \(-2\) and \(-3\) are greater than \(-4\).
- The final answer is \(\boxed{\{-2, -3\}}\).
Explanation
- Problem Analysis and Objective
We are given the set \(S = \{-2, -3, -4, -5\}\) and the inequality \(2p - 8 < 5p + 4\). Our objective is to identify which integers from the set \(S\) satisfy this inequality. To do this, we will first simplify the inequality to isolate the variable \(p\).
- Simplifying the Inequality
Starting with the given inequality:
\[2p - 8 < 5p + 4\]
We subtract \(2p\) from both sides to group the variable terms:
\[-8 < 5p - 2p + 4\]
\[-8 < 3p + 4\]
Next, we subtract 4 from both sides:
\[-8 - 4 < 3p\]
\[-12 < 3p\]
Finally, we divide both sides by 3 to solve for \(p\):
\[\frac{-12}{3} < p\]
\[-4 < p\]
This means the inequality \(2p - 8 < 5p + 4\) is equivalent to \(p > -4\).
- Testing the Elements of Set S
Now, we test each integer \(p\) in the set \(S = \{-2, -3, -4, -5\}\) against the condition \(p > -4\):
- For \(p = -2\): \(-2 > -4\) is true.
- For \(p = -3\): \(-3 > -4\) is true.
- For \(p = -4\): \(-4 > -4\) is false.
- For \(p = -5\): \(-5 > -4\) is false.
The integers from the set \(S\) that satisfy the inequality are \(-2\) and \(-3\).
- Conclusion
Based on our analysis, the integers from the set \(\{-2, -3, -4, -5\}\) that satisfy the inequality \(2p - 8 < 5p + 4\) are \(-2\) and \(-3\). Thus, the correct choice is \(\{-2, -3\}\).
Examples
In real life, inequalities help define boundaries or conditions for success. For example, if a company wants to ensure that the production cost of an item \(p\) (where \(p\) represents some index or variable) is optimized such that \(2p - 8 < 5p + 4\), they can use this math to determine the valid range of parameters to maintain their budget or operational goals. By identifying which conditions satisfy their constraints, they can make informed decisions in resource management.