Which point justifies the shaded area by satisfying $\frac{(y+1)^2}{6}-\frac{(x+2)^2}{24} \geq 1$?
A. $(12,6)$
B. $(-2,6)$
C. $(8,-6)$
D. $(-1,-2)$
Answer (1)
Substitute each point into the inequality.
Check if the inequality holds true for each point.
Point ( − 2 , 6 ) satisfies the inequality: 6 ( 6 + 1 ) 2 − 24 ( − 2 + 2 ) 2 = 6 49 ≥ 1 .
The point that satisfies the inequality is ( − 2 , 6 ) .
Explanation
Understanding the Problem We are given the inequality 6 ( y + 1 ) 2 − 24 ( x + 2 ) 2 ≥ 1 and four points: ( 12 , 6 ) , ( − 2 , 6 ) , ( 8 , − 6 ) , and ( − 1 , − 2 ) . We need to determine which point satisfies the inequality.
Solution Plan We will substitute each point into the inequality and check if the inequality holds true.
Substituting the Points
Point ( 12 , 6 ) : Substituting x = 12 and y = 6 into the inequality, we get 6 ( 6 + 1 ) 2 − 24 ( 12 + 2 ) 2 = 6 49 − 24 196 = 6 49 − 6 49 = 0 Since 0 ≥ 1 is false, the point ( 12 , 6 ) does not satisfy the inequality.
Point ( − 2 , 6 ) : Substituting x = − 2 and y = 6 into the inequality, we get 6 ( 6 + 1 ) 2 − 24 ( − 2 + 2 ) 2 = 6 49 − 24 0 = 6 49 = 8.1666... Since 8.1666... ≥ 1 is true, the point ( − 2 , 6 ) satisfies the inequality.
Point ( 8 , − 6 ) : Substituting x = 8 and y = − 6 into the inequality, we get 6 ( − 6 + 1 ) 2 − 24 ( 8 + 2 ) 2 = 6 25 − 24 100 = 6 25 − 6 25 = 0 Since 0 ≥ 1 is false, the point ( 8 , − 6 ) does not satisfy the inequality.
Point ( − 1 , − 2 ) : Substituting x = − 1 and y = − 2 into the inequality, we get 6 ( − 2 + 1 ) 2 − 24 ( − 1 + 2 ) 2 = 6 1 − 24 1 = 24 4 − 24 1 = 24 3 = 8 1 = 0.125 Since 0.125 ≥ 1 is false, the point ( − 1 , − 2 ) does not satisfy the inequality.
Conclusion Therefore, only the point ( − 2 , 6 ) satisfies the inequality 6 ( y + 1 ) 2 − 24 ( x + 2 ) 2 ≥ 1 .
Examples
This type of inequality can represent the feasible region in optimization problems. For example, a company might want to determine the optimal production levels of two products, where x and y represent the quantities of each product. The inequality could represent a constraint on resources or demand, and the company would want to find production levels (x, y) that satisfy the constraint and maximize profit. In this case, the point (-2, 6) would represent a feasible production level that meets the specified constraint.
