Which of the following ordered pairs is a solution of the system of inequalities?
[tex]$\begin{array}{l}
y\ \textgreater \ 2 x^2-8 \
y \leq-x^2-3 x+4\n\end{array}$[/tex]
A) [tex]$(5,0)$[/tex]
B) [tex]$(-3,-2)$[/tex]
C) [tex]$(4,-3)$[/tex]
Answer (2)
Test each ordered pair in both inequalities.
(5,0) fails the first inequality: 2(5)^2 - 8"> 0 > 2 ( 5 ) 2 − 8 is false.
(-3,-2) fails the first inequality: 2(-3)^2 - 8"> − 2 > 2 ( − 3 ) 2 − 8 is false.
(4,-3) fails the first inequality: 2(4)^2 - 8"> − 3 > 2 ( 4 ) 2 − 8 is false.
Therefore, none of the given ordered pairs are solutions to the system. There is no correct answer among the options.
Explanation
Analyze the problem We are given a system of two inequalities:
2x^2 - 8"> y > 2 x 2 − 8
and
$y
We need to check which of the given ordered pairs (5,0), (-3,-2), and (4,-3) satisfies both inequalities.
Test (5,0) Let's test the first ordered pair (5,0). Substitute x = 5 and y = 0 into the inequalities:
For the first inequality:
2(5)^2 - 8"> 0 > 2 ( 5 ) 2 − 8
2(25) - 8"> 0 > 2 ( 25 ) − 8
50 - 8"> 0 > 50 − 8
42"> 0 > 42 . This is false.
Since the first inequality is false, the ordered pair (5,0) is not a solution to the system of inequalities.
Test (-3,-2) Now let's test the second ordered pair (-3,-2). Substitute x = -3 and y = -2 into the inequalities:
For the first inequality:
2(-3)^2 - 8"> − 2 > 2 ( − 3 ) 2 − 8
2(9) - 8"> − 2 > 2 ( 9 ) − 8
18 - 8"> − 2 > 18 − 8
10"> − 2 > 10 . This is false.
Since the first inequality is false, the ordered pair (-3,-2) is not a solution to the system of inequalities.
Test (4,-3) Now let's test the third ordered pair (4,-3). Substitute x = 4 and y = -3 into the inequalities:
For the first inequality:
2(4)^2 - 8"> − 3 > 2 ( 4 ) 2 − 8
2(16) - 8"> − 3 > 2 ( 16 ) − 8
32 - 8"> − 3 > 32 − 8
24"> − 3 > 24 . This is false.
Since the first inequality is false, the ordered pair (4,-3) is not a solution to the system of inequalities.
Conclusion Since none of the ordered pairs satisfy the first inequality, none of them are solutions to the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as optimization problems in economics and engineering. For example, a company might use a system of inequalities to determine the optimal production levels of two different products, given constraints on resources like labor and materials. Each inequality represents a constraint, and the solution set represents the feasible region where all constraints are satisfied. By finding the corner points of this region, the company can determine the production levels that maximize profit while staying within the resource constraints.
None of the ordered pairs satisfy both inequalities of the system. Therefore, none of the options (5,0), (-3,-2), or (4,-3) are solutions to the inequalities given. There is no correct answer among the provided choices.
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