The sum of two positive integers, [tex]$x$[/tex] and [tex]$y$[/tex], is not more than 40. The difference of the two integers is at least 20. Haneece chooses [tex]$x$[/tex] as the larger number and uses the inequalities [tex]$y \leq 40-x$[/tex] and [tex]$y \leq x-20$[/tex] to determine the possible solutions. She determines that [tex]$x$[/tex] must be between 0 and 10 and [tex]$y$[/tex] must be between 20 and 40. Determine if Haneece found the correct solution. If not, state the correct solution.
A. Yes, Chaneece found the correct solution.
B. No, Chaneece mixed up the variables. The correct solution is that [tex]$x$[/tex] must be between 20 and 40 and [tex]$y$[/tex] must be between 0 and 10.
C. No, Chaneece should not have restricted the solution to quadrant I. The correct solution is that [tex]$x$[/tex] can be all real numbers and [tex]$y$[/tex] must be less than 10.
D. No, Chancece looked at the wrong area of the shaded graph. The correct solution is that [tex]$x$[/tex] must be between 0 and 30 and [tex]$y$[/tex] must be between 0 and 40.
Answer (2)
Analyze the given inequalities: x + y ≤ 40 and x − y ≥ 20 .
Find the intersection point by setting 40 − x = x − 20 , which gives x = 30 .
Determine the bounds for x and y : 21 ≤ x ≤ 30 and 1 ≤ y ≤ 10 .
Conclude that Haneece's solution is incorrect, and the correct solution is 21 ≤ x ≤ 30 and 1 ≤ y ≤ 10 .
The correct solution is that x must be between 21 and 30 and y must be between 1 and 10. The answer is: N o , C han eece mi x e d u pt h e v a r iab l es . T h ecorrec t so l u t i o ni s t ha t x m u s t b e b e tw ee n 21 an d 30 an d y m u s t b e b e tw ee n 1 an d 10.
Explanation
Problem Analysis Let's analyze the problem. We are given that x and y are positive integers such that their sum is not more than 40, which means x + y ≤ 40 . We are also given that their difference is at least 20, and x is the larger number, so x − y ≥ 20 . Haneece uses the inequalities y ≤ 40 − x and y ≤ x − 20 and concludes that 0 ≤ x ≤ 10 and 20 ≤ y ≤ 40 . We need to determine if Haneece's solution is correct.
Given Inequalities From the given inequalities, we have:
x + y ≤ 40 , which implies y ≤ 40 − x .
x − y ≥ 20 , which implies y ≤ x − 20 .
Since x and y are positive integers, 0"> x > 0 and 0"> y > 0 .
Finding the Intersection To find the intersection of the inequalities, we set 40 − x = x − 20 , which gives 2 x = 60 , so x = 30 . This means that when x = 30 , y = 40 − 30 = 10 and y = 30 − 20 = 10 . So the two inequalities intersect at x = 30 and y = 10 .
Finding the Bounds Since 0"> y > 0 , we have 0"> x − 20 > 0 , so 20"> x > 20 . Also, 0"> 40 − x > 0 , so x < 40 . Thus, 20 < x < 40 .
Adding the inequalities x + y ≤ 40 and x − y ≥ 20 , we get 2 x ≤ 60 , so x ≤ 30 . Subtracting the inequalities, we get 2 y ≤ 20 , so y ≤ 10 .
Refining the Bounds Since x − y ≥ 20 and 0"> y > 0 , we have 20"> x > 20 . Since x + y ≤ 40 and 0"> y > 0 , we have x < 40 . Therefore, 20 < x < 40 .
Since x − y ≥ 20 , we have y ≤ x − 20 . Since x + y ≤ 40 , we have y ≤ 40 − x . Thus, y ≤ min ( x − 20 , 40 − x ) .
The Correct Solution Since x and y are positive integers, x ≥ 21 and y ≥ 1 . Also, x ≤ 30 and y ≤ 10 . Therefore, 21 ≤ x ≤ 30 and 1 ≤ y ≤ 10 .
Comparing with Haneece's Solution Comparing this with Haneece's solution, we see that Haneece's solution is incorrect. The correct solution is that 21 ≤ x ≤ 30 and 1 ≤ y ≤ 10 .
Final Answer Therefore, Haneece's solution is incorrect. The correct solution is that x must be between 21 and 30, and y must be between 1 and 10.
Examples
Understanding inequalities helps in various real-life situations. For example, when planning a budget, you might have constraints on how much you can spend on different categories (like food, rent, and entertainment). These constraints can be expressed as inequalities. Similarly, in manufacturing, companies use inequalities to ensure that their products meet certain quality standards while minimizing costs. Inequalities are also used in optimization problems, such as finding the most efficient way to allocate resources or schedule tasks.
Haneece's conclusion was incorrect; the correct ranges are that x must be between 21 and 39 while y must equal 1.
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