\(\begin{array}{l}
y>-3 x+3 \\
y \geq 2 x-2
\end{array}\)
(1,0)
(-1,1)
(2,2)
(0,3)
Answer (1)
Substitute each point into the inequalities.
Check if both inequalities are true for each point.
The point (2, 2) satisfies both inequalities.
The solution is ( 2 , 2 ) .
Explanation
Problem Analysis We are given a system of inequalities and a set of points. Our goal is to determine which of the given points satisfy both inequalities. The inequalities are:
-3x + 3"> y > − 3 x + 3 = 2x - 2"> y " >= 2 x − 2
The points are: ( 1 , 0 ) , ( − 1 , 1 ) , ( 2 , 2 ) , ( 0 , 3 ) . We will substitute each point into both inequalities to see if they hold true.
Testing Point (1, 0) Let's test the point ( 1 , 0 ) .
For the first inequality: -3(1) + 3"> 0 > − 3 ( 1 ) + 3 -3 + 3"> 0 > − 3 + 3 0"> 0 > 0 This is false.
For the second inequality: = 2(1) - 2"> 0" >= 2 ( 1 ) − 2 = 2 - 2"> 0" >= 2 − 2 = 0"> 0" >= 0 This is true.
Since the first inequality is false, the point ( 1 , 0 ) does not satisfy the system of inequalities.
Testing Point (-1, 1) Let's test the point ( − 1 , 1 ) .
For the first inequality: -3(-1) + 3"> 1 > − 3 ( − 1 ) + 3 3 + 3"> 1 > 3 + 3 6"> 1 > 6 This is false.
For the second inequality: = 2(-1) - 2"> 1" >= 2 ( − 1 ) − 2 = -2 - 2"> 1" >= − 2 − 2 = -4"> 1" >= − 4 This is true.
Since the first inequality is false, the point ( − 1 , 1 ) does not satisfy the system of inequalities.
Testing Point (2, 2) Let's test the point ( 2 , 2 ) .
For the first inequality: -3(2) + 3"> 2 > − 3 ( 2 ) + 3 -6 + 3"> 2 > − 6 + 3 -3"> 2 > − 3 This is true.
For the second inequality: = 2(2) - 2"> 2" >= 2 ( 2 ) − 2 = 4 - 2"> 2" >= 4 − 2 = 2"> 2" >= 2 This is true.
Since both inequalities are true, the point ( 2 , 2 ) satisfies the system of inequalities.
Testing Point (0, 3) Let's test the point ( 0 , 3 ) .
For the first inequality: -3(0) + 3"> 3 > − 3 ( 0 ) + 3 0 + 3"> 3 > 0 + 3 3"> 3 > 3 This is false.
For the second inequality: = 2(0) - 2"> 3" >= 2 ( 0 ) − 2 = 0 - 2"> 3" >= 0 − 2 = -2"> 3" >= − 2 This is true.
Since the first inequality is false, the point ( 0 , 3 ) does not satisfy the system of inequalities.
Conclusion Only the point ( 2 , 2 ) satisfies both inequalities. Therefore, the solution is ( 2 , 2 ) .
Examples
Systems of inequalities are used in various real-world applications, such as optimizing resource allocation, determining feasible regions in linear programming, and modeling constraints in engineering design. For instance, a company might use a system of inequalities to determine the optimal production levels of different products, given constraints on available resources like labor and materials. By identifying the region that satisfies all constraints, the company can make informed decisions to maximize profit or minimize costs.
