Which system of linear inequalities has the point $(2,1)$ in its solution set?
$\begin{array}{l}
y<-x+3 \\
y \leq \frac{1}{2} x+3
\end{array}$
Answer (1)
Substitute x = 2 and y = 1 into the first inequality: 1 < − 2 + 3 , which simplifies to 1 < 1 . This is false.
Substitute x = 2 and y = 1 into the second inequality: 1 ≤ 2 1 ( 2 ) + 3 , which simplifies to 1 ≤ 4 . This is true.
Since the first inequality is false, the point ( 2 , 1 ) is not in the solution set.
Therefore, the point ( 2 , 1 ) is not in the solution set of the given system of inequalities. F a l se
Explanation
Checking the point (2,1) We are given the system of inequalities:
y < − x + 3 y
We need to check if the point ( 2 , 1 ) is in the solution set of this system. This means we need to substitute x = 2 and y = 1 into both inequalities and see if they are both true.
First Inequality Let's substitute x = 2 and y = 1 into the first inequality:
y < − x + 3
1 < − 2 + 3
1 < 1
This inequality is false, since 1 is not less than 1.
Second Inequality Now let's substitute x = 2 and y = 1 into the second inequality:
$y
$1
$1
$1
This inequality is true, since 1 is less than or equal to 4.
Conclusion Since the first inequality is false when we substitute x = 2 and y = 1 , the point ( 2 , 1 ) is not in the solution set of the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where they help determine the optimal solution for a problem with constraints. For example, a company might use a system of inequalities to determine the optimal production levels of different products, given constraints on resources like labor and materials. By graphing the inequalities, the company can visualize the feasible region and find the production levels that maximize profit while satisfying all constraints. This approach ensures efficient resource allocation and informed decision-making.
