$\begin{array}{l}
y<5 x+2 \\
y \geq \frac{1}{2} x+1
\end{array}$
Is (-1,3), (0,2), (1,2), (2,-1), or (2,2) a solution?
Answer (1)
Substitute each point into both inequalities.
Check if both inequalities are true for that point.
The point ( 1 , 2 ) satisfies both inequalities: 2 < 5 ( 1 ) + 2 and 2 ≥ 2 1 ( 1 ) + 1 .
The point ( 2 , 2 ) satisfies both inequalities: 2 < 5 ( 2 ) + 2 and 2 ≥ 2 1 ( 2 ) + 1 .
( 1 , 2 ) , ( 2 , 2 )
Explanation
Problem Analysis We are given a system of inequalities and a set of points. Our goal is to determine which points satisfy both inequalities. The inequalities are:
y < 5 x + 2 = \frac{1}{2}x + 1"> y " >= 2 1 x + 1
The points are: ( − 1 , 3 ) , ( 0 , 2 ) , ( 1 , 2 ) , ( 2 , − 1 ) , ( 2 , 2 ) . We will substitute each point into both inequalities to see if they hold true.
Testing (-1, 3) Let's test the point ( − 1 , 3 ) :
For the first inequality: 3 < 5 ( − 1 ) + 2 3 < − 5 + 2 3 < − 3 This is false.
Since the first inequality is false, the point ( − 1 , 3 ) does not satisfy the system of inequalities.
Testing (0, 2) Let's test the point ( 0 , 2 ) :
For the first inequality: 2 < 5 ( 0 ) + 2 2 < 0 + 2 2 < 2 This is false.
Since the first inequality is false, the point ( 0 , 2 ) does not satisfy the system of inequalities.
Testing (1, 2) Let's test the point ( 1 , 2 ) :
For the first inequality: 2 < 5 ( 1 ) + 2 2 < 5 + 2 2 < 7 This is true.
For the second inequality: 2 ≥ 2 1 ( 1 ) + 1 2 ≥ 2 1 + 1 2 ≥ 2 3 This is true.
Since both inequalities are true, the point ( 1 , 2 ) satisfies the system of inequalities.
Testing (2, -1) Let's test the point ( 2 , − 1 ) :
For the first inequality: − 1 < 5 ( 2 ) + 2 − 1 < 10 + 2 − 1 < 12 This is true.
For the second inequality: − 1 ≥ 2 1 ( 2 ) + 1 − 1 ≥ 1 + 1 − 1 ≥ 2 This is false.
Since the second inequality is false, the point ( 2 , − 1 ) does not satisfy the system of inequalities.
Testing (2, 2) Let's test the point ( 2 , 2 ) :
For the first inequality: 2 < 5 ( 2 ) + 2 2 < 10 + 2 2 < 12 This is true.
For the second inequality: 2 ≥ 2 1 ( 2 ) + 1 2 ≥ 1 + 1 2 ≥ 2 This is true.
Since both inequalities are true, the point ( 2 , 2 ) satisfies the system of inequalities.
Final Answer Therefore, the points that satisfy the system of inequalities are ( 1 , 2 ) and ( 2 , 2 ) .
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, to optimize solutions under constraints. For example, a company might use a system of inequalities to determine the optimal production levels of two products, given constraints on resources like labor and materials. Similarly, diet planning can use inequalities to ensure nutritional requirements are met within certain calorie limits. These mathematical tools help in making informed decisions in resource allocation and optimization problems.
