Which point is a solution to the linear inequality [tex]y\ \textless \ -\frac{1}{2} x[/tex] +2 ?
A. (2,3)
B. (2,1)
C. (3,-2)
D. (-1,3)
Answer (1)
Substitute each point into the inequality y < − 2 1 x + 2 .
Check if the inequality holds true for each point.
( 2 , 3 ) : 3 < − 2 1 ( 2 ) + 2 ⇒ 3 < 1 (False)
( 2 , 1 ) : 1 < − 2 1 ( 2 ) + 2 ⇒ 1 < 1 (False)
( 3 , − 2 ) : − 2 < − 2 1 ( 3 ) + 2 ⇒ − 2 < 0.5 (True)
( − 1 , 3 ) : 3 < − 2 1 ( − 1 ) + 2 ⇒ 3 < 2.5 (False)
The only point that satisfies the inequality is ( 3 , − 2 ) .
Explanation
Understanding the Problem We are given the inequality y < − 2 1 x + 2 and four points: ( 2 , 3 ) , ( 2 , 1 ) , ( 3 , − 2 ) , and ( − 1 , 3 ) . We need to determine which of these points satisfy the inequality.
Testing the Points Let's test each point to see if it satisfies the inequality.
Testing (2,3) For the point ( 2 , 3 ) , we substitute x = 2 and y = 3 into the inequality: 3 < − 2 1 ( 2 ) + 2 3 < − 1 + 2 3 < 1 This is false, so ( 2 , 3 ) is not a solution.
Testing (2,1) For the point ( 2 , 1 ) , we substitute x = 2 and y = 1 into the inequality: 1 < − 2 1 ( 2 ) + 2 1 < − 1 + 2 1 < 1 This is false, so ( 2 , 1 ) is not a solution.
Testing (3,-2) For the point ( 3 , − 2 ) , we substitute x = 3 and y = − 2 into the inequality: − 2 < − 2 1 ( 3 ) + 2 − 2 < − 2 3 + 2 − 2 < − 1.5 + 2 − 2 < 0.5 This is true, so ( 3 , − 2 ) is a solution.
Testing (-1,3) For the point ( − 1 , 3 ) , we substitute x = − 1 and y = 3 into the inequality: 3 < − 2 1 ( − 1 ) + 2 3 < 2 1 + 2 3 < 2.5 This is false, so ( − 1 , 3 ) is not a solution.
Conclusion Therefore, the only point that satisfies the inequality is ( 3 , − 2 ) .
Examples
Linear inequalities are used in various real-world scenarios, such as determining if a certain budget constraint is met. For example, if you have a budget of $20 and want to buy apples and bananas, where apples cost $2 each and bananas cost $1 each, the inequality 2 x + y < 20 represents the combinations of apples (x) and bananas (y) you can buy without exceeding your budget. Checking if a specific combination, like 5 apples and 8 bananas, satisfies the inequality helps you determine if it's a feasible purchase within your budget.
