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$125 x+y$
$12.2 x+y$
$12 \leq 2 x+3 y$
$12>2 x+3 y$
Answer (1)
Rewrite the inequalities as 2 x + 3 y ≥ 12 and 2 x + 3 y < 12 .
Analyze the line 2 x + 3 y = 12 , which passes through ( 0 , 4 ) and ( 6 , 0 ) .
The region 2 x + 3 y ≥ 12 is above or on the line.
The region 2 x + 3 y < 12 is below the line.
The expressions 125 x + y and 12.2 x + y are not used due to the problem being ill-defined.
Explanation
Understanding the Problem We are given the expressions 125 x + y and 12.2 x + y , and the inequalities 12 ≤ 2 x + 3 y and 2x + 3y"> 12 > 2 x + 3 y . The problem is not well-defined, so we will assume that the objective is to find the region defined by the inequalities 12 ≤ 2 x + 3 y and 2x + 3y"> 12 > 2 x + 3 y .
Rewriting the Inequalities The inequalities 12 ≤ 2 x + 3 y and 2x + 3y"> 12 > 2 x + 3 y define two regions in the x y -plane. We can rewrite the inequalities as 2 x + 3 y ≥ 12 and 2 x + 3 y < 12 .
Analyzing the Line Let's analyze the line 2 x + 3 y = 12 . We can find the intercepts by setting x = 0 and y = 0 . If x = 0 , then 3 y = 12 , so y = 4 . If y = 0 , then 2 x = 12 , so x = 6 . Thus, the line passes through the points ( 0 , 4 ) and ( 6 , 0 ) .
Region for the First Inequality For 2 x + 3 y ≥ 12 , the region is above or on the line 2 x + 3 y = 12 . This includes the line itself.
Region for the Second Inequality For 2 x + 3 y < 12 , the region is below the line 2 x + 3 y = 12 . This does not include the line itself.
Conclusion The expressions 125 x + y and 12.2 x + y might be related to optimization, but without a clear objective function, it is impossible to proceed further. The inequalities define two distinct regions separated by the line 2 x + 3 y = 12 .
Examples
Imagine you're planning a party and need to stay within a budget. The inequalities can represent constraints on the number of items you can buy, like snacks and drinks. For example, 2 x + 3 y ≥ 12 could mean you need to buy enough snacks ( x ) and drinks ( y ) to serve at least 12 guests, where each snack costs $2 and each drink costs $3. Understanding these inequalities helps you make informed decisions to stay within your budget while ensuring everyone has enough to enjoy.
