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
\end{array}
[/tex]
A) $(5,0)$
B) $(-3,-2)$
C) $(4,-3)$
D) $(0,-5)$
Answer (1)
Test each ordered pair in both inequalities.
(5,0) fails the first inequality.
(-3,-2) fails the first inequality.
(4,-3) fails the first inequality.
(0,-5) satisfies both inequalities.
The solution is ( 0 , − 5 ) .
Explanation
Analyze the problem We are given a system of two inequalities:
2x^2 - 8"> y > 2 x 2 − 8
$y
We need to check which of the given ordered pairs (5,0), (-3,-2), (4,-3), and (0,-5) satisfies both inequalities.
Test (5,0) Let's test the first ordered pair (5,0). Substitute x = 5 and y = 0 into the inequalities:
2(5)^2 - 8 \Rightarrow 0 > 2(25) - 8 \Rightarrow 0 > 50 - 8 \Rightarrow 0 > 42"> 0 > 2 ( 5 ) 2 − 8 ⇒ 0 > 2 ( 25 ) − 8 ⇒ 0 > 50 − 8 ⇒ 0 > 42 (False)
$0
Since the first inequality is false, (5,0) is not a solution.
Test (-3,-2) Now let's test the second ordered pair (-3,-2). Substitute x = -3 and y = -2 into the inequalities:
2(-3)^2 - 8 \Rightarrow -2 > 2(9) - 8 \Rightarrow -2 > 18 - 8 \Rightarrow -2 > 10"> − 2 > 2 ( − 3 ) 2 − 8 ⇒ − 2 > 2 ( 9 ) − 8 ⇒ − 2 > 18 − 8 ⇒ − 2 > 10 (False)
$-2
Since the first inequality is false, (-3,-2) is not a solution.
Test (4,-3) Next, let's test the third ordered pair (4,-3). Substitute x = 4 and y = -3 into the inequalities:
2(4)^2 - 8 \Rightarrow -3 > 2(16) - 8 \Rightarrow -3 > 32 - 8 \Rightarrow -3 > 24"> − 3 > 2 ( 4 ) 2 − 8 ⇒ − 3 > 2 ( 16 ) − 8 ⇒ − 3 > 32 − 8 ⇒ − 3 > 24 (False)
$-3
Since the first inequality is false, (4,-3) is not a solution.
Test (0,-5) Finally, let's test the fourth ordered pair (0,-5). Substitute x = 0 and y = -5 into the inequalities:
2(0)^2 - 8 \Rightarrow -5 > 0 - 8 \Rightarrow -5 > -8"> − 5 > 2 ( 0 ) 2 − 8 ⇒ − 5 > 0 − 8 ⇒ − 5 > − 8 (True)
$-5
Since both inequalities are true, (0,-5) is a solution.
Conclusion Therefore, the ordered pair (0,-5) is a solution to the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as optimization problems in business and economics. 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. The solution to the system would represent the range of production levels that satisfy all the constraints and maximize profit. Similarly, in nutrition, systems of inequalities can help determine the range of food quantities that meet specific dietary requirements while staying within a certain budget.
