Explanation Given information: The system of inequalities is, { x + y ≤ 12 y ≤ 1 2 x − 6 y ≤ 2 x + 6 Graph: The given system of the inequalities is, x + y ≤ 12 ...... (1) y ≤ 1 2 x − 6 ...... (2) y ≤ 2 x + 6 ...... (3) The corresponding equation of the inequality (1) is, x + y = 12 Since, the inequality x + y ≤ 12 involves the less than or equal to sign, then the points on the line x + y = 12 satisfies the inequality x + y ≤ 12 . Therefore, the line itself is the part of the solution set. The corresponding equation of the inequality (2) is, y = 1 2 x − 6 Since, the inequality y ≤ 1 2 x − 6 involves the less than or equal to sign, then the points on the line y = 1 2 x − 6 satisfies the inequality y ≤ 1 2 x − 6 . Therefore, the line y = 1 2 x − 6 itself is the part of the solution set. The corresponding equation of the inequality (3) is, y = 2 x + 6 Since, the inequality y ≤ 2 x + 6 involves the less than or equal to sign, then the points on the line y = 2 x + 6 satisfies the inequality y ≤ 2 x + 6 . Therefore, the line y = 2 x + 6 itself is the part of the solution set. Consider the test point ( 0 , − 8 ) to check whether the solution satisfies each inequality of the given system or not. Substitute 0 for x and − 8 for y in the inequality x + y ≤ 12 . x + y ≤ ? 12 0 + ( − 8 ) ≤ ? 12 − 8 ≤ 12 The point ( 0 , − 8 ) lies below the line x + y = 12 and satisfies the inequality x + y ≤ 12 . So it is the part of the solution set. Substitute 0 for x and − 8 for y in the inequality y ≤ 1 2 x − 6 . y ≤ ? 1 2 x − 6 − 8 ≤ ? 1 2 ( 0 ) − 6 − 8 ≤ − 6 The point ( 0 , − 8 ) lies below the line y = 1 2 x − 6 and satisfies the inequality y ≤ 1 2 x − 6 . So it is the part of the solution set. Substitute 0 for x and − 8 for y in the inequality y ≤ 2 x + 6 . y ≤ ? 2 x + 6 − 8 ≤ ? 2 ( 0 ) + 6 − 8 ≤ 6 The point ( 0 , − 8 ) lies below the line y = 2 x + 6 and satisfies the inequality y ≤ 2 x + 6 . So it is the part of the solution set. Therefore, the test point ( 0 , − 8 ) satisfies each inequality of the given system. The solution set of the given system of inequalities is the intersection of the solutions of each of the given inequality. Therefore, the solution set is shown as shaded region in Figure 1. The vertices occur at the points of intersection of the corresponding equation of the given system of inequalities. x + 2 y = 12 ...... (4) y = 1 2 x − 6

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 5.5, Problem 61E

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The system of inequality {x+y≤12y≤12x−6y≤2x+6. Also find the coordinates of all vertices, and check whether the solution set is bounded

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Chapter 5.5, Problem 60E

Chapter 5.5, Problem 62E

Chapter 5 Solutions

College Algebra

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To graph: The system of inequality { x + y ≤ 12 y ≤ 1 2 x − 6 y ≤ 2 x + 6 . Also find the coordinates of all vertices, and check whether the solution set is bounded

Solution Summary: The author graphs the system of inequality lx+y

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To graph:

The system of inequality {x+y≤12y≤12x−6y≤2x+6. Also find the coordinates of all vertices, and check whether the solution set is bounded

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Chapter 5 Solutions

To graph: The system of inequality { x + y ≤ 12 y ≤ 1 2 x − 6 y ≤ 2 x + 6 . Also find the coordinates of all vertices, and check whether the solution set is bounded

Solution Summary: The author graphs the system of inequality lx+y

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To graph: The system of inequality {x+y≤12y≤12x−6y≤2x+6. Also find the coordinates of all vertices, and check whether the solution set is bounded

Want to see the full answer?

Chapter 5 Solutions

To graph:

The system of inequality {x+y≤12y≤12x−6y≤2x+6. Also find the coordinates of all vertices, and check whether the solution set is bounded