$\begin{array}{l}
y>-3 x+3 \\
y \geq 2 x-2
\end{array}$
Answer (1)
Substitute each point into the inequalities.
Check if both inequalities are true for each point.
( 1 , 0 ) : -3(1) + 3"> 0 > − 3 ( 1 ) + 3 is false, so ( 1 , 0 ) is not a solution.
( − 1 , 1 ) : -3(-1) + 3"> 1 > − 3 ( − 1 ) + 3 is false, so ( − 1 , 1 ) is not a solution.
( 2 , 2 ) : -3(2) + 3"> 2 > − 3 ( 2 ) + 3 and = "2(2) - 2"> 2" >= "2 ( 2 ) − 2 are both true, so ( 2 , 2 ) is a solution.
The point that satisfies both inequalities is ( 2 , 2 ) .
Explanation
Understanding the Problem We are given two inequalities: -3x + 3"> y > − 3 x + 3 and = "2x - 2"> y " >= "2 x − 2 . We need to check which of the given points ( 1 , 0 ) , ( − 1 , 1 ) , and ( 2 , 2 ) satisfy both inequalities.
Checking Point (1, 0) Let's check the point ( 1 , 0 ) .
For the first inequality, -3x + 3"> y > − 3 x + 3 , we have -3(1) + 3"> 0 > − 3 ( 1 ) + 3 , which simplifies to 0"> 0 > 0 . This is false. For the second inequality, = "2x - 2"> y " >= "2 x − 2 , we have = "2(1) - 2"> 0" >= "2 ( 1 ) − 2 , which simplifies to = "0"> 0" >= "0 . This is true. Since the first inequality is not satisfied, the point ( 1 , 0 ) does not satisfy both inequalities.
Checking Point (-1, 1) Now let's check the point ( − 1 , 1 ) .
For the first inequality, -3x + 3"> y > − 3 x + 3 , we have -3(-1) + 3"> 1 > − 3 ( − 1 ) + 3 , which simplifies to 3 + 3"> 1 > 3 + 3 , or 6"> 1 > 6 . This is false. For the second inequality, = "2x - 2"> y " >= "2 x − 2 , we have = "2(-1) - 2"> 1" >= "2 ( − 1 ) − 2 , which simplifies to = "-2 - 2"> 1" >= " − 2 − 2 , or = "-4"> 1" >= " − 4 . This is true. Since the first inequality is not satisfied, the point ( − 1 , 1 ) does not satisfy both inequalities.
Checking Point (2, 2) Finally, let's check the point ( 2 , 2 ) .
For the first inequality, -3x + 3"> y > − 3 x + 3 , we have -3(2) + 3"> 2 > − 3 ( 2 ) + 3 , which simplifies to -6 + 3"> 2 > − 6 + 3 , or -3"> 2 > − 3 . This is true. For the second inequality, = "2x - 2"> y " >= "2 x − 2 , we have = "2(2) - 2"> 2" >= "2 ( 2 ) − 2 , which simplifies to = "4 - 2"> 2" >= "4 − 2 , or = "2"> 2" >= "2 . This is true. Since both inequalities are satisfied, the point ( 2 , 2 ) satisfies both inequalities.
Conclusion Therefore, only the point ( 2 , 2 ) satisfies both inequalities.
Examples
Understanding inequalities helps in various real-life scenarios, such as budgeting, where you need to ensure your expenses are less than or equal to your income. Similarly, in manufacturing, quality control involves ensuring that product dimensions fall within specified tolerances. In resource allocation, inequalities help determine the optimal distribution of resources to maximize efficiency while adhering to constraints.
