Given $4 y^2+x-32 y+68>0$, which ordered pair is part of the solution set?
A. $(-3,5.7)$
B. $(-5.9,4)$
C. $(-8,5)$
D. $(-13,3)$
Answer (1)
Substitute each ordered pair into the inequality.
Check if the inequality holds true.
The ordered pair ( − 3 , 5.7 ) satisfies the inequality: 0"> 4 ( 5.7 ) 2 + ( − 3 ) − 32 ( 5.7 ) + 68 = 12.56 > 0 .
Therefore, the ordered pair that is part of the solution set is ( − 3 , 5.7 ) .
Explanation
Problem Analysis We are given the inequality 0"> 4 y 2 + x − 32 y + 68 > 0 and four ordered pairs: ( − 3 , 5.7 ) , ( − 5.9 , 4 ) , ( − 8 , 5 ) , and ( − 13 , 3 ) . We need to determine which of these pairs satisfies the inequality. We will substitute the x and y values from each pair into the inequality and check if the result is greater than 0.
Testing (-3, 5.7) Let's test the first ordered pair ( − 3 , 5.7 ) . Substituting x = − 3 and y = 5.7 into the inequality, we get: 0"> 4 ( 5.7 ) 2 + ( − 3 ) − 32 ( 5.7 ) + 68 > 0 0"> 4 ( 32.49 ) − 3 − 182.4 + 68 > 0 0"> 129.96 − 3 − 182.4 + 68 > 0 0"> 12.56 > 0 Since 0"> 12.56 > 0 , the first ordered pair ( − 3 , 5.7 ) satisfies the inequality.
Testing (-5.9, 4) Let's test the second ordered pair ( − 5.9 , 4 ) . Substituting x = − 5.9 and y = 4 into the inequality, we get: 0"> 4 ( 4 ) 2 + ( − 5.9 ) − 32 ( 4 ) + 68 > 0 0"> 4 ( 16 ) − 5.9 − 128 + 68 > 0 0"> 64 − 5.9 − 128 + 68 > 0 0"> − 1.9 > 0 Since − 1.9 is not greater than 0, the second ordered pair ( − 5.9 , 4 ) does not satisfy the inequality.
Testing (-8, 5) Let's test the third ordered pair ( − 8 , 5 ) . Substituting x = − 8 and y = 5 into the inequality, we get: 0"> 4 ( 5 ) 2 + ( − 8 ) − 32 ( 5 ) + 68 > 0 0"> 4 ( 25 ) − 8 − 160 + 68 > 0 0"> 100 − 8 − 160 + 68 > 0 0"> 0 > 0 Since 0 is not greater than 0, the third ordered pair ( − 8 , 5 ) does not satisfy the inequality.
Testing (-13, 3) Let's test the fourth ordered pair ( − 13 , 3 ) . Substituting x = − 13 and y = 3 into the inequality, we get: 0"> 4 ( 3 ) 2 + ( − 13 ) − 32 ( 3 ) + 68 > 0 0"> 4 ( 9 ) − 13 − 96 + 68 > 0 0"> 36 − 13 − 96 + 68 > 0 0"> − 5 > 0 Since − 5 is not greater than 0, the fourth ordered pair ( − 13 , 3 ) does not satisfy the inequality.
Conclusion Only the first ordered pair ( − 3 , 5.7 ) satisfies the inequality 0"> 4 y 2 + x − 32 y + 68 > 0 .
Examples
Understanding inequalities helps in various real-world scenarios, such as determining if a certain combination of resources meets a specific requirement. For example, a chef might use inequalities to ensure that a dish contains enough nutrients while staying within a certain calorie limit. Similarly, an engineer could use inequalities to verify that a structure can withstand a certain amount of stress without failing. In finance, inequalities can help determine if an investment strategy will yield a desired return while minimizing risk. These applications highlight the practical importance of understanding and solving inequalities.
