What is the solution to the inequality [tex]$28>4-2 p$[/tex]?
A. {p | p>-12}
B. {p | p<-12}
C. {p | p>12}
D. {p | p<12}
Answer (2)
Subtract 4 from both sides of the inequality: -2p"> 24 > − 2 p .
Divide both sides by -2, flipping the inequality sign: − 12 < p .
Rewrite the inequality: -12"> p > − 12 .
Express the solution set: -12}}"> p ∣ p > − 12 .
Explanation
Understanding the Inequality We are given the inequality 4 - 2p"> 28 > 4 − 2 p . Our goal is to isolate p to find the solution set.
Subtracting 4 from Both Sides First, we subtract 4 from both sides of the inequality to simplify it: 4 - 2p - 4"> 28 − 4 > 4 − 2 p − 4 -2p"> 24 > − 2 p
Dividing by -2 and Flipping the Inequality Next, we divide both sides by -2. Remember that when we divide by a negative number, we must flip the inequality sign: − 2 24 < − 2 − 2 p − 12 < p
Expressing the Solution Set This inequality − 12 < p is equivalent to -12"> p > − 12 . Therefore, the solution set is all p such that p is greater than -12. In set notation, this is written as -12}"> p ∣ p > − 12 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, imagine you're managing a budget and need to ensure your expenses don't exceed your income. If your income is represented by a fixed number and your expenses are related to a variable, an inequality helps you determine the maximum value that variable can take without breaking your budget. Similarly, in science, inequalities are used to define acceptable ranges for experimental conditions, ensuring results remain valid and safe. Inequalities also play a vital role in optimization problems, helping to find the best possible outcome within given constraints.
The solution to the inequality 4 - 2p"> 28 > 4 − 2 p is -12"> p > − 12 . Therefore, the correct choice is option A: {p | p > -12}. This means that any value of p greater than -12 satisfies the inequality.
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