Which system of linear inequalities has the point $(3,-2)$ in its solution set?
$\begin{array}{l}
y<-3 \\
y \leq \frac{2}{3} x-4
\end{array}$
Answer (1)
Substitute the point ( 3 , − 2 ) into the first inequality y < − 3 , which results in − 2 < − 3 , which is false.
Substitute the point ( 3 , − 2 ) into the second inequality y ≤ 3 2 x − 4 , which results in − 2 ≤ − 2 , which is true.
Since the point must satisfy both inequalities to be a solution, and the first inequality is not satisfied, the point ( 3 , − 2 ) is not a solution to the system.
Therefore, the point ( 3 , − 2 ) is not in the solution set of the given system of inequalities.
Explanation
Problem Analysis and Setup We are given the point ( 3 , − 2 ) and the system of inequalities:
y < − 3 y ≤ 3 2 x − 4
We need to determine if the point ( 3 , − 2 ) is a solution to this system. To do this, we will substitute x = 3 and y = − 2 into each inequality and check if the inequalities hold true.
Checking the First Inequality First, let's substitute x = 3 and y = − 2 into the first inequality, y < − 3 :
− 2 < − 3
This statement is false, since − 2 is greater than − 3 .
Checking the Second Inequality Now, let's substitute x = 3 and y = − 2 into the second inequality, y ≤ 3 2 x − 4 :
− 2 ≤ 3 2 ( 3 ) − 4 − 2 ≤ 2 − 4 − 2 ≤ − 2
This statement is true, since − 2 is equal to − 2 .
Conclusion For the point ( 3 , − 2 ) to be a solution to the system of inequalities, both inequalities must be true. However, the first inequality is false, and the second inequality is true. Therefore, the point ( 3 , − 2 ) is not a solution to the given system of inequalities.
Final Answer Since the point ( 3 , − 2 ) does not satisfy the first inequality, it is not in the solution set of the system of inequalities.
Examples
Systems of inequalities are used in many 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. By graphing the inequalities, the company can find the feasible region, representing all possible production levels that satisfy the constraints. The optimal solution, maximizing profit, can then be found within this region.
