Which ordered pair makes both inequalities true?
[tex]
\begin{array}{l}
y>-3 x+3 \\
y \geq 2 x-2
\end{array}
[/tex]
A. $(1,0)$
B. $(-1,1)$
C. $(2,2)$
D. $(0,3)$
Answer (2)
The ordered pair that makes both inequalities true is (2,2). We verified each option and found that (2,2) is the only one satisfying both inequalities. Therefore, the answer is (2,2).
;
Test each ordered pair in both inequalities.
(1,0): -3(1) + 3"> 0 > − 3 ( 1 ) + 3 is false.
(-1,1): -3(-1) + 3"> 1 > − 3 ( − 1 ) + 3 is false.
(2,2): -3(2) + 3"> 2 > − 3 ( 2 ) + 3 and = "2(2) - 2"> 2" >= "2 ( 2 ) − 2 are both true.
(0,3): -3(0) + 3"> 3 > − 3 ( 0 ) + 3 is false.
The ordered pair that makes both inequalities true is ( 2 , 2 ) .
Explanation
Analyze the problem We are given two inequalities: -3x + 3"> y > − 3 x + 3 and = "2x - 2"> y " >= "2 x − 2 . We need to find which of the given ordered pairs satisfies both inequalities. Let's test each ordered pair.
Test (1,0) Let's test the ordered pair (1,0). For the first inequality, we have -3(1) + 3"> 0 > − 3 ( 1 ) + 3 , which simplifies to 0"> 0 > 0 . This is false. For the second inequality, we have = "2(1) - 2"> 0" >= "2 ( 1 ) − 2 , which simplifies to = "0"> 0" >= "0 . This is true. Since the first inequality is false, (1,0) is not a solution.
Test (-1,1) Let's test the ordered pair (-1,1). For the first inequality, 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, 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 false, (-1,1) is not a solution.
Test (2,2) Let's test the ordered pair (2,2). For the first inequality, 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, 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 true, (2,2) is a solution.
Test (0,3) Let's test the ordered pair (0,3). For the first inequality, we have -3(0) + 3"> 3 > − 3 ( 0 ) + 3 , which simplifies to 0 + 3"> 3 > 0 + 3 , or 3"> 3 > 3 . This is false. For the second inequality, we have = "2(0) - 2"> 3" >= "2 ( 0 ) − 2 , which simplifies to = "0 - 2"> 3" >= "0 − 2 , or = "-2"> 3" >= " − 2 . This is true. Since the first inequality is false, (0,3) is not a solution.
Conclusion The only ordered pair that satisfies both inequalities is (2,2).
Examples
Understanding inequalities is crucial in various real-life scenarios, such as budgeting and resource allocation. For instance, if you have a limited budget and need to decide how much to spend on different items while ensuring you stay within your budget and meet certain minimum requirements, inequalities can help you model and solve such problems. Similarly, in manufacturing, inequalities can be used to optimize production processes while adhering to constraints on resources and quality standards.
