Which points are solutions to the linear inequality [tex]y\ \textless \ 0.5x+2[/tex]? Select three options.
[tex](-3,-2)[/tex]
[tex](-2,1)[/tex]
[tex](-1,-2)[/tex]
[tex](-1,2)[/tex]
[tex](1,-2)[/tex]
Answer (1)
Substitute each point into the inequality y < 0.5 x + 2 .
Check if the inequality holds true for each point.
The inequality holds true for ( − 3 , − 2 ) because − 2 < 0.5 ( − 3 ) + 2 = 0.5 .
The inequality holds true for ( − 1 , − 2 ) because − 2 < 0.5 ( − 1 ) + 2 = 1.5 .
The inequality holds true for ( 1 , − 2 ) because − 2 < 0.5 ( 1 ) + 2 = 2.5 .
The points that satisfy the inequality are ( − 3 , − 2 ) , ( − 1 , − 2 ) , and ( 1 , − 2 ) . Thus, the answer is ( − 3 , − 2 ) , ( − 1 , − 2 ) , ( 1 , − 2 ) .
Explanation
Understanding the Problem We are given the inequality y < 0.5 x + 2 and five points: ( − 3 , − 2 ) , ( − 2 , 1 ) , ( − 1 , − 2 ) , ( − 1 , 2 ) , and ( 1 , − 2 ) . We need to determine which three points satisfy the inequality.
Solution Strategy For each point, we will substitute the x and y values into the inequality and check if it holds true.
Checking Point 1 Point 1: ( − 3 , − 2 ) . Substituting x = − 3 and y = − 2 into the inequality, we get: − 2 < 0.5 ( − 3 ) + 2
− 2 < − 1.5 + 2 − 2 < 0.5 This is true.
Checking Point 2 Point 2: ( − 2 , 1 ) . Substituting x = − 2 and y = 1 into the inequality, we get: 1 < 0.5 ( − 2 ) + 2 1 < − 1 + 2 1 < 1 This is false.
Checking Point 3 Point 3: ( − 1 , − 2 ) . Substituting x = − 1 and y = − 2 into the inequality, we get: − 2 < 0.5 ( − 1 ) + 2 − 2 < − 0.5 + 2 − 2 < 1.5 This is true.
Checking Point 4 Point 4: ( − 1 , 2 ) . Substituting x = − 1 and y = 2 into the inequality, we get: 2 < 0.5 ( − 1 ) + 2 2 < − 0.5 + 2 2 < 1.5 This is false.
Checking Point 5 Point 5: ( 1 , − 2 ) . Substituting x = 1 and y = − 2 into the inequality, we get: − 2 < 0.5 ( 1 ) + 2 − 2 < 0.5 + 2 − 2 < 2.5 This is true.
Final Answer The points that satisfy the inequality are ( − 3 , − 2 ) , ( − 1 , − 2 ) , and ( 1 , − 2 ) .
Examples
Linear inequalities are used in various real-world scenarios, such as determining budget constraints. For example, if you have a budget of $50 and want to buy apples and bananas, where apples cost $2 each and bananas cost $1 each, the inequality 2 x + y < 50 represents the possible combinations of apples ( x ) and bananas ( y ) you can buy within your budget. Similarly, in business, companies use linear inequalities to model production constraints, resource allocation, and profit maximization.
