For what values of $p$ is the following statement true?
Square $A$ has a perimeter that is less than the perimeter of rectangle $B$.
Inequality
$\qquad$ < $2($ $\qquad$ $)+2(p+$ $\qquad$ $)$
Solution
$\qquad$ < $3+2 p+$ $\qquad$
$\qquad$ < $2 p+$
$\qquad$ $\qquad$ < $\qquad$
$p<$
$\qquad$
$p$ is $\qquad$ than $\qquad$ units
Answer (2)
The inequality derived indicates that for the perimeter of square A to be less than that of rectangle B , p must be greater than 0. Therefore, the solution is 0"> p > 0 units. This means any positive value for p will satisfy the relationship between the perimeters.
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Assume the perimeter of square A is 3, leading to the inequality 3 < 2 ( ) + 2 ( p + ) .
Complete the inequality with the given information: 3 < 3 + 2 p .
Simplify the inequality: 0 < 2 p .
Solve for p : 0"> p > 0 , meaning p is greater than 0 units. The final answer is 0}"> p > 0 .
Explanation
Problem Analysis Let's analyze the given problem. We are given an incomplete inequality that relates the perimeter of a square A to the perimeter of a rectangle B . Our goal is to find the values of p for which the perimeter of square A is less than the perimeter of rectangle B . We need to complete the inequality and solve for p .
Completing the Inequality and Solving for p The given inequality is:
< 2 ( ) + 2 ( p + ) .
From the solution steps, we have:
< 3 + 2 p +
< 2 p +
<
p <
p is than units
Let's assume that the missing terms in the inequality are constants. From the first line of the solution, we can infer that the perimeter of square A is a constant. Let's assume the perimeter of square A is 3. Then the inequality becomes:
3 < 2 ( ) + 2 ( p + ) .
From the next line, we have:
3 < 3 + 2 p +
This implies that the missing term is 0. So, the inequality becomes:
3 < 3 + 2 p
Subtracting 3 from both sides, we get:
0 < 2 p
Dividing both sides by 2, we get:
0 < p
So, 0"> p > 0 .
Now, let's fill in the missing terms in the solution:
3 < 3 + 2 p + 0
3 < 3 + 2 p
0 < 2 p
0 < p
0"> p > 0
p is greater than 0 units.
Alternative Scenario Now, let's consider a slightly different scenario where the initial inequality is: 3 < 2 ( 1/2 ) + 2 ( p + 1/2 ) .
Then, 3 < 1 + 2 p + 1 , which simplifies to 3 < 2 + 2 p . Subtracting 2 from both sides, we get 1 < 2 p , so 1/2 = 0.5"> p > 1/2 = 0.5 .
However, based on the provided solution steps, it seems the intention is that the perimeter of square A is 3, and the inequality simplifies as follows:
3 < 3 + 2 p 0 < 2 p 0 < p 0"> p > 0
Final Solution Based on the given information, the completed inequality and solution are:
3 < 3 + 2 p 0 < 2 p 0 < p 0"> p > 0
Therefore, p is greater than 0 units.
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when planning a budget, you might want to ensure that your expenses are less than your income. Similarly, in cooking, you might need to adjust ingredient quantities to maintain a certain flavor profile. In manufacturing, ensuring that the dimensions of a product fall within a specific range is essential for quality control. These situations all involve working with inequalities to achieve desired outcomes.
