Which ordered pair makes both inequalities true?
$\begin{array}{l}
y>-3 x+3 \\
y \geq 2 x-2
\end{array}$
A. $(1,0)$
B. $(-1,1)$
C. $(2,2)$
D. $(0,3)$
Answer (1)
Test each ordered pair in both inequalities.
( 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.
( 0 , 3 ) : -3(0) + 3"> 3 > − 3 ( 0 ) + 3 is false, so ( 0 , 3 ) is not a solution.
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, -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 false, the ordered pair ( 1 , 0 ) does not satisfy both inequalities.
Test (-1,1) Let's test the ordered pair ( − 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 , so 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 , so = -4"> 1" >= − 4 . This is true. Since the first inequality is false, the ordered pair ( − 1 , 1 ) does not satisfy both inequalities.
Test (2,2) Let's test the ordered pair ( 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 , so -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 , so = 2"> 2" >= 2 . This is true. Since both inequalities are true, the ordered pair ( 2 , 2 ) satisfies both inequalities.
Test (0,3) Let's test the ordered pair ( 0 , 3 ) .
For the first inequality, -3x + 3"> y > − 3 x + 3 , we have -3(0) + 3"> 3 > − 3 ( 0 ) + 3 , which simplifies to 0 + 3"> 3 > 0 + 3 , so 3"> 3 > 3 . This is false. For the second inequality, = 2x - 2"> y " >= 2 x − 2 , we have = 2(0) - 2"> 3" >= 2 ( 0 ) − 2 , which simplifies to = 0 - 2"> 3" >= 0 − 2 , so = -2"> 3" >= − 2 . This is true. Since the first inequality is false, the ordered pair ( 0 , 3 ) does not satisfy both inequalities.
Conclusion The only ordered pair that satisfies both inequalities is ( 2 , 2 ) .
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 cooking, you might need to keep the amount of ingredients within a certain range to achieve the desired taste. Inequalities are also used in setting speed limits on roads, ensuring vehicles travel at a safe speed.
