Which ordered pairs are in the solution set of the system of linear inequalities?
[tex]
\begin{array}{l}
y \geq-\frac{1}{2} x \\
y\ \textless \ \frac{1}{2} x+1
\end{array}
[/tex]
(5,-2), (3,1), (-4,2)
(5,-2), (3,-1), (4,-3)
(5,-2), (3,1), (4,2)
(5,-2), (-3,1), (4,2)
Answer (1)
Test each ordered pair in the given sets against both inequalities.
The ordered pair (x, y) must satisfy y ≥ − 2 1 x and y < 2 1 x + 1 .
Eliminate sets where at least one ordered pair does not satisfy both inequalities.
The solution set is ( 5 , − 2 ) , ( 3 , 1 ) , ( 4 , 2 ) .
Explanation
Understanding the Problem We are given a system of two linear inequalities:
y ≥ − 2 1 x
y < 2 1 x + 1
We need to check which of the given sets of ordered pairs satisfy both inequalities.
Testing the First Set of Ordered Pairs Let's test the first set of ordered pairs: ( 5 , − 2 ) , ( 3 , 1 ) , ( − 4 , 2 ) .
For ( 5 , − 2 ) :
− 2 ≥ − 2 1 ( 5 ) = − 2.5 This is true since -2.5"> − 2 > − 2.5 .
− 2 < 2 1 ( 5 ) + 1 = 2.5 + 1 = 3.5 This is true since − 2 < 3.5 .
For ( 3 , 1 ) :
1 ≥ − 2 1 ( 3 ) = − 1.5 This is true since -1.5"> 1 > − 1.5 .
1 < 2 1 ( 3 ) + 1 = 1.5 + 1 = 2.5 This is true since 1 < 2.5 .
For ( − 4 , 2 ) :
2 ≥ − 2 1 ( − 4 ) = 2 This is true since 2 = 2 .
2 < 2 1 ( − 4 ) + 1 = − 2 + 1 = − 1 This is false since -1"> 2 > − 1 .
Since ( − 4 , 2 ) does not satisfy both inequalities, the first set is not the solution.
Testing the Second Set of Ordered Pairs Let's test the second set of ordered pairs: ( 5 , − 2 ) , ( 3 , − 1 ) , ( 4 , − 3 ) .
For ( 5 , − 2 ) :
− 2 ≥ − 2 1 ( 5 ) = − 2.5 This is true since -2.5"> − 2 > − 2.5 .
− 2 < 2 1 ( 5 ) + 1 = 2.5 + 1 = 3.5 This is true since − 2 < 3.5 .
For ( 3 , − 1 ) :
− 1 ≥ − 2 1 ( 3 ) = − 1.5 This is true since -1.5"> − 1 > − 1.5 .
− 1 < 2 1 ( 3 ) + 1 = 1.5 + 1 = 2.5 This is true since − 1 < 2.5 .
For ( 4 , − 3 ) :
− 3 ≥ − 2 1 ( 4 ) = − 2 This is false since − 3 < − 2 .
Since ( 4 , − 3 ) does not satisfy both inequalities, the second set is not the solution.
Testing the Third Set of Ordered Pairs Let's test the third set of ordered pairs: ( 5 , − 2 ) , ( 3 , 1 ) , ( 4 , 2 ) .
For ( 5 , − 2 ) :
− 2 ≥ − 2 1 ( 5 ) = − 2.5 This is true since -2.5"> − 2 > − 2.5 .
− 2 < 2 1 ( 5 ) + 1 = 2.5 + 1 = 3.5 This is true since − 2 < 3.5 .
For ( 3 , 1 ) :
1 ≥ − 2 1 ( 3 ) = − 1.5 This is true since -1.5"> 1 > − 1.5 .
1 < 2 1 ( 3 ) + 1 = 1.5 + 1 = 2.5 This is true since 1 < 2.5 .
For ( 4 , 2 ) :
2 ≥ − 2 1 ( 4 ) = − 2 This is true since -2"> 2 > − 2 .
2 < 2 1 ( 4 ) + 1 = 2 + 1 = 3 This is true since 2 < 3 .
Since all ordered pairs in this set satisfy both inequalities, this is the solution.
Testing the Fourth Set of Ordered Pairs Let's test the fourth set of ordered pairs: ( 5 , − 2 ) , ( − 3 , 1 ) , ( 4 , 2 ) .
For ( 5 , − 2 ) :
− 2 ≥ − 2 1 ( 5 ) = − 2.5 This is true since -2.5"> − 2 > − 2.5 .
− 2 < 2 1 ( 5 ) + 1 = 2.5 + 1 = 3.5 This is true since − 2 < 3.5 .
For ( − 3 , 1 ) :
1 ≥ − 2 1 ( − 3 ) = 1.5 This is false since 1 < 1.5 .
Since ( − 3 , 1 ) does not satisfy both inequalities, the fourth set is not the solution.
Final Answer The ordered pairs in the solution set of the system of linear inequalities are ( 5 , − 2 ) , ( 3 , 1 ) , ( 4 , 2 ) .
Examples
Understanding systems of inequalities is crucial in various real-world applications. For instance, consider a scenario where a company needs to optimize its production. They might have constraints on resources like labor hours and raw materials. Each constraint can be represented as an inequality, and the solution set of the system of inequalities would represent the feasible production plans that satisfy all constraints. By identifying the solution set, the company can make informed decisions about production levels to maximize profit while adhering to resource limitations. This approach is fundamental in fields like operations research and economics, where optimization problems are frequently encountered.
