What ordered pair is the solution to the following system of linear inequalities?
[tex]2 x+3 y\ \textless \ 5[/tex]
[tex]3 x-7 y \geq 0[/tex]
A. (-2,4)
B. (3,0)
C. (0,2)
D. (1,-1)
Answer (1)
Test each ordered pair in the inequalities.
The ordered pair ( − 2 , 4 ) does not satisfy the first inequality.
The ordered pair ( 3 , 0 ) does not satisfy the first inequality.
The ordered pair ( 0 , 2 ) does not satisfy the first inequality.
The ordered pair ( 1 , − 1 ) satisfies both inequalities, so the solution is ( 1 , − 1 ) .
Explanation
Problem Analysis We are given a system of two linear inequalities and four possible solutions. Our task is to find which of the given ordered pairs satisfies both inequalities. We will test each ordered pair to see if it satisfies both inequalities.
Testing (-2, 4) Let's test the first ordered pair ( − 2 , 4 ) in the inequalities:
Inequality 1: 2 x + 3 y < 5 Substitute x = − 2 and y = 4 : 2 ( − 2 ) + 3 ( 4 ) < 5 ⇒ − 4 + 12 < 5 ⇒ 8 < 5 . This is false.
Since the first inequality is not satisfied, we don't need to check the second inequality.
Testing (3, 0) Let's test the second ordered pair ( 3 , 0 ) in the inequalities:
Inequality 1: 2 x + 3 y < 5 Substitute x = 3 and y = 0 : 2 ( 3 ) + 3 ( 0 ) < 5 ⇒ 6 + 0 < 5 ⇒ 6 < 5 . This is false.
Since the first inequality is not satisfied, we don't need to check the second inequality.
Testing (0, 2) Let's test the third ordered pair ( 0 , 2 ) in the inequalities:
Inequality 1: 2 x + 3 y < 5 Substitute x = 0 and y = 2 : 2 ( 0 ) + 3 ( 2 ) < 5 ⇒ 0 + 6 < 5 ⇒ 6 < 5 . This is false.
Since the first inequality is not satisfied, we don't need to check the second inequality.
Testing (1, -1) Let's test the fourth ordered pair ( 1 , − 1 ) in the inequalities:
Inequality 1: 2 x + 3 y < 5 Substitute x = 1 and y = − 1 : 2 ( 1 ) + 3 ( − 1 ) < 5 ⇒ 2 − 3 < 5 ⇒ − 1 < 5 . This is true.
Inequality 2: 3 x − 7 y ≥ 0 Substitute x = 1 and y = − 1 : 3 ( 1 ) − 7 ( − 1 ) ≥ 0 ⇒ 3 + 7 ≥ 0 ⇒ 10 ≥ 0 . This is true.
Since both inequalities are satisfied, the ordered pair ( 1 , − 1 ) is the solution.
Conclusion Therefore, the ordered pair that satisfies both inequalities is ( 1 , − 1 ) .
Examples
Systems of inequalities are used in various real-world applications, such as in economics to determine feasible production regions given resource constraints, in diet planning to find combinations of foods that meet nutritional requirements within budget limits, and in engineering to design structures that satisfy multiple safety and performance criteria. For instance, a company might use a system of inequalities to determine the optimal number of products to manufacture, given constraints on labor, materials, and demand. By identifying the region of feasible solutions, the company can make informed decisions that maximize profit while adhering to resource limitations.
