An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Test the first set of points (2,2), (3,1), (4,2) and verify that all points satisfy both inequalities.
Test the second set of points (2,2), (3,-1), (4,1) and find that (3,-1) does not satisfy the first inequality.
Test the third set of points (2,2), (1,-2), (0,2) and find that (1,-2) does not satisfy the first inequality.
Test the fourth set of points (2,2), (1,2), (2,0) and find that (2,0) does not satisfy the first inequality. The solution is therefore: ( 2 , 2 ) , ( 3 , 1 ) , ( 4 , 2 )
Explanation
Problem Analysis We are given two inequalities: y ≥ − 3 1 x + 2 y < 2 x + 3 We need to determine which of the given sets of points satisfies both inequalities.
Testing the First Set of Points Let's test the first set of points (2,2), (3,1), (4,2): For (2,2): 2 ≥ − 3 1 ( 2 ) + 2 = − 3 2 + 2 = 3 4 , which is true. 2 < 2 ( 2 ) + 3 = 4 + 3 = 7 , which is true. For (3,1): 1 ≥ − 3 1 ( 3 ) + 2 = − 1 + 2 = 1 , which is true. 1 < 2 ( 3 ) + 3 = 6 + 3 = 9 , which is true. For (4,2): 2 ≥ − 3 1 ( 4 ) + 2 = − 3 4 + 2 = 3 2 , which is true. 2 < 2 ( 4 ) + 3 = 8 + 3 = 11 , which is true. Since all points in the first set satisfy both inequalities, this set is a potential solution.
Testing the Second Set of Points Let's test the second set of points (2,2), (3,-1), (4,1): For (2,2): 2 ≥ − 3 1 ( 2 ) + 2 = 3 4 , which is true. 2 < 2 ( 2 ) + 3 = 7 , which is true. For (3,-1): − 1 ≥ − 3 1 ( 3 ) + 2 = − 1 + 2 = 1 , which is false. Since one inequality is false, we don't need to check the second inequality. Since not all points in the second set satisfy both inequalities, this set is not a solution.
Testing the Third Set of Points Let's test the third set of points (2,2), (1,-2), (0,2): For (2,2): 2 ≥ − 3 1 ( 2 ) + 2 = 3 4 , which is true. 2 < 2 ( 2 ) + 3 = 7 , which is true. For (1,-2): − 2 ≥ − 3 1 ( 1 ) + 2 = − 3 1 + 2 = 3 5 , which is false. Since one inequality is false, we don't need to check the second inequality. Since not all points in the third set satisfy both inequalities, this set is not a solution.
Testing the Fourth Set of Points Let's test the fourth set of points (2,2), (1,2), (2,0): For (2,2): 2 ≥ − 3 1 ( 2 ) + 2 = 3 4 , which is true. 2 < 2 ( 2 ) + 3 = 7 , which is true. For (1,2): 2 ≥ − 3 1 ( 1 ) + 2 = − 3 1 + 2 = 3 5 , which is true. 2 < 2 ( 1 ) + 3 = 2 + 3 = 5 , which is true. For (2,0): 0 ≥ − 3 1 ( 2 ) + 2 = − 3 2 + 2 = 3 4 , which is false. Since one inequality is false, we don't need to check the second inequality. Since not all points in the fourth set satisfy both inequalities, this set is not a solution.
Conclusion Therefore, only the first set of points (2,2), (3,1), (4,2) satisfies both inequalities.
Examples
Understanding systems of inequalities is crucial in various real-world applications, such as optimizing resource allocation. For instance, a company might use inequalities to determine the optimal production levels of different products, considering constraints like available materials, labor hours, and market demand. By identifying the feasible region that satisfies all constraints, the company can make informed decisions to maximize profit or minimize costs. This approach ensures efficient use of resources and strategic planning in complex scenarios.
