Which point is a solution to the following system of inequalities: $4 x+2 y<8$ and $x-6 y<-2$?
A. $(3,6)$
B. $(0,0)$
C. $(1,1)$
D. $(-6,-1)$
Answer (1)
Test each point in the inequalities.
Point ( 3 , 6 ) : 4 ( 3 ) + 2 ( 6 ) = 24 < 8 is false.
Point ( 0 , 0 ) : 4 ( 0 ) + 2 ( 0 ) = 0 < 8 is true, but 0 − 6 ( 0 ) = 0 < − 2 is false.
Point ( 1 , 1 ) : 4 ( 1 ) + 2 ( 1 ) = 6 < 8 is true, and 1 − 6 ( 1 ) = − 5 < − 2 is true.
Point ( − 6 , − 1 ) : 4 ( − 6 ) + 2 ( − 1 ) = − 26 < 8 is true, but − 6 − 6 ( − 1 ) = 0 < − 2 is false.
Therefore, the solution is ( 1 , 1 ) .
Explanation
Analyze the problem We are given a system of two inequalities: 4 x + 2 y < 8 and x − 6 y < − 2 . We need to find which of the given points satisfies both inequalities. Let's test each point.
Test point (3, 6) Let's test the point ( 3 , 6 ) .
For the first inequality, 4 x + 2 y < 8 , we have: 4 ( 3 ) + 2 ( 6 ) = 12 + 12 = 24 . Since 24 < 8 is false, ( 3 , 6 ) is not a solution.
Test point (0, 0) Let's test the point ( 0 , 0 ) .
For the first inequality, 4 x + 2 y < 8 , we have: 4 ( 0 ) + 2 ( 0 ) = 0 . Since 0 < 8 is true, the first inequality is satisfied.
For the second inequality, x − 6 y < − 2 , we have: 0 − 6 ( 0 ) = 0 . Since 0 < − 2 is false, ( 0 , 0 ) is not a solution.
Test point (1, 1) Let's test the point ( 1 , 1 ) .
For the first inequality, 4 x + 2 y < 8 , we have: 4 ( 1 ) + 2 ( 1 ) = 4 + 2 = 6 . Since 6 < 8 is true, the first inequality is satisfied.
For the second inequality, x − 6 y < − 2 , we have: 1 − 6 ( 1 ) = 1 − 6 = − 5 . Since − 5 < − 2 is true, the second inequality is satisfied.
Therefore, ( 1 , 1 ) is a solution to the system of inequalities.
Test point (-6, -1) Let's test the point ( − 6 , − 1 ) .
For the first inequality, 4 x + 2 y < 8 , we have: 4 ( − 6 ) + 2 ( − 1 ) = − 24 − 2 = − 26 . Since − 26 < 8 is true, the first inequality is satisfied.
For the second inequality, x − 6 y < − 2 , we have: − 6 − 6 ( − 1 ) = − 6 + 6 = 0 . Since 0 < − 2 is false, ( − 6 , − 1 ) is not a solution.
Final Answer The point ( 1 , 1 ) satisfies both inequalities. Therefore, ( 1 , 1 ) is a solution to the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where you want to optimize a certain objective function subject to constraints. For example, a company might want to maximize its profit given constraints on resources like labor and materials. Each constraint can be represented as an inequality, and the solution to the system of inequalities represents the feasible region where all constraints are satisfied. In this case, we found a point that satisfies both inequalities, which means it lies within the feasible region.
