Solve the compound inequality below.
[tex]$-3 \leq \frac{p}{2}\ \textless \ 0$[/tex]
Answer (2)
Multiply all parts of the inequality by 2: − 6 ≤ p < 0 .
Express the solution: p ≥ − 6 and p < 0 .
Compare the solution with the given options.
The correct solution is: p ≥ − 6 and p < 0
Explanation
Understanding the Inequality We are given the compound inequality − 3 ≤ 2 p < 0 . Our goal is to isolate p to find the solution set.
Multiplying by 2 To isolate p , we need to eliminate the fraction. We can do this by multiplying all parts of the inequality by 2. This gives us: 2 × ( − 3 ) ≤ 2 × 2 p < 2 × 0
Simplifying Simplifying the inequality, we get: − 6 ≤ p < 0
Interpreting the Solution This inequality states that p is greater than or equal to -6 and less than 0. In other words, p must be in the interval [ − 6 , 0 ) . Now, let's compare this solution to the given options to find the correct one.
Comparing with Options The given options are:
p < − 6 or 2"> p > 2
p < − 6 or 0"> p > 0
p ≥ − 6 and p < 0
all real numbers
Option 1 is incorrect because it includes values less than -6 and greater than 2, which are not in our solution set. Option 2 is incorrect for the same reason as option 1. Option 3, p ≥ − 6 and p < 0 , is equivalent to − 6 ≤ p < 0 , which matches our solution. Option 4, all real numbers, is incorrect because our solution is a specific interval, not all real numbers.
Final Answer Therefore, the correct solution is p ≥ − 6 and p < 0 .
Examples
Imagine you're baking a cake and the recipe says the oven temperature should be between -3 and 0 (in some unit, let's say a special baking scale). If your oven's scale is twice the standard, you need to adjust the recipe's temperature range. Solving the inequality − 3 ≤ 2 p < 0 helps you find the correct temperature range p for your oven, which turns out to be between -6 and 0 on the standard scale. This ensures your cake bakes perfectly!
To solve the compound inequality − 3 ≤ 2 p < 0 , multiply by 2 to get − 6 ≤ p < 0 . Thus, the solution is p ≥ − 6 and p < 0 . This can also be expressed in interval notation as [ − 6 , 0 ) .
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