Which of the following ordered pairs is not a solution of the inequality $y>x^2+4x+3$?
(2,17)
(1,8)
(0,5)
(-3,4)
Answer (2)
The problem asks us to identify which ordered pair does not satisfy the inequality x^2 + 4x + 3"> y > x 2 + 4 x + 3 .
We substitute each ordered pair into the inequality and check if the inequality holds true.
For ( 2 , 17 ) , 2^2 + 4(2) + 3 \Rightarrow 17 > 15"> 17 > 2 2 + 4 ( 2 ) + 3 ⇒ 17 > 15 , which is true.
For ( 1 , 8 ) , 1^2 + 4(1) + 3 \Rightarrow 8 > 8"> 8 > 1 2 + 4 ( 1 ) + 3 ⇒ 8 > 8 , which is false. Therefore, the answer is ( 1 , 8 ) .
Explanation
Understanding the Problem We are given the inequality x^2 + 4x + 3"> y > x 2 + 4 x + 3 and four ordered pairs: ( 2 , 17 ) , ( 1 , 8 ) , ( 0 , 5 ) , and ( − 3 , 4 ) . We need to find the ordered pair that does not satisfy the inequality. This means we need to substitute the x and y values of each ordered pair into the inequality and check if the inequality holds true. If it doesn't, then that ordered pair is not a solution.
Testing Each Ordered Pair Let's test each ordered pair:
( 2 , 17 ) : Substitute x = 2 and y = 17 into the inequality: (2)^2 + 4(2) + 3"> 17 > ( 2 ) 2 + 4 ( 2 ) + 3 4 + 8 + 3"> 17 > 4 + 8 + 3 15"> 17 > 15 . This is true, so ( 2 , 17 ) is a solution.
( 1 , 8 ) : Substitute x = 1 and y = 8 into the inequality: (1)^2 + 4(1) + 3"> 8 > ( 1 ) 2 + 4 ( 1 ) + 3 1 + 4 + 3"> 8 > 1 + 4 + 3 8"> 8 > 8 . This is false, so ( 1 , 8 ) is not a solution.
( 0 , 5 ) : Substitute x = 0 and y = 5 into the inequality: (0)^2 + 4(0) + 3"> 5 > ( 0 ) 2 + 4 ( 0 ) + 3 0 + 0 + 3"> 5 > 0 + 0 + 3 3"> 5 > 3 . This is true, so ( 0 , 5 ) is a solution.
( − 3 , 4 ) : Substitute x = − 3 and y = 4 into the inequality: (-3)^2 + 4(-3) + 3"> 4 > ( − 3 ) 2 + 4 ( − 3 ) + 3 9 - 12 + 3"> 4 > 9 − 12 + 3 0"> 4 > 0 . This is true, so ( − 3 , 4 ) is a solution.
Finding the Non-Solution After testing each ordered pair, we found that only ( 1 , 8 ) does not satisfy the inequality x^2 + 4x + 3"> y > x 2 + 4 x + 3 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when designing a bridge, engineers use inequalities to ensure the structure can withstand a certain load. Similarly, in economics, inequalities help determine price ranges that maximize profit while keeping costs within budget. In everyday life, inequalities are used to compare deals and make informed decisions about spending and saving.
The ordered pair that is not a solution of the inequality x^2 + 4x + 3"> y > x 2 + 4 x + 3 is ( 1 , 8 ) because it does not satisfy the inequality when substituted into it.
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