Is $(1,3)$ a solution to the system of inequalities below?
[tex]\begin{array}{l}
y\ \textgreater \ 3x-1 \
y\ \textless \ -3 x
\end{array}[/tex]
Why or why not?
A. Yes, $(1,3)$ is not a solution to the inequalities because when you plug in the 1 for $x$ and the 3 for $y$ it creates two true statements.
B. Yes, $(1,3)$ is not a solution to the inequalities because when you plug in the 1 for $x$ and the 3 for $y$ it creates two false statements.
C. No, $(1,3)$ is not a solution to the inequalities because when you plug in the 1 for $x$ and the 3 for $y$ it creates two false statements.
D. No, $(1,3)$ is not a solution to the inequalities because when you plug in the 1 for $x$ and the 3 for $y$...
Answer (2)
The point ( 1 , 3 ) is not a solution to the given system of inequalities because it satisfies the first inequality but not the second. Therefore, the answer is No. We conclude that the correct choice is option D.
;
Substitute x = 1 and y = 3 into the first inequality 3x - 1"> y > 3 x − 1 , which yields 3(1) - 1"> 3 > 3 ( 1 ) − 1 , simplifying to 2"> 3 > 2 . This is true.
Substitute x = 1 and y = 3 into the second inequality y < − 3 x , which yields 3 < − 3 ( 1 ) , simplifying to 3 < − 3 . This is false.
Since one inequality is true and the other is false, the point ( 1 , 3 ) is not a solution to the system of inequalities.
Therefore, the answer is No, ( 1 , 3 ) is not a solution. $\boxed{No}
Explanation
Analyze the problem We are given the system of inequalities:
3x - 1"> y > 3 x − 1 y < − 3 x
We need to determine if the point ( 1 , 3 ) is a solution to this system. To do this, we will substitute x = 1 and y = 3 into both inequalities and check if the resulting statements are true.
Check the first inequality First, let's substitute x = 1 and y = 3 into the first inequality:
3x - 1"> y > 3 x − 1 3(1) - 1"> 3 > 3 ( 1 ) − 1 3 - 1"> 3 > 3 − 1 2"> 3 > 2
This statement is true.
Check the second inequality Now, let's substitute x = 1 and y = 3 into the second inequality:
y < − 3 x 3 < − 3 ( 1 ) 3 < − 3
This statement is false.
Conclusion Since the point ( 1 , 3 ) satisfies the first inequality but not the second inequality, it is not a solution to the system of inequalities. Therefore, the correct answer is:
No, ( 1 , 3 ) is not a solution to the inequalities because when you plug in the 1 for x and the 3 for y it creates one true and one false statement.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where you want to optimize a certain objective function subject to constraints. For example, a company might want to maximize its profit given constraints on the amount of resources available, such as labor, materials, and equipment. By formulating the problem as a system of inequalities, the company can find the optimal solution that satisfies all the constraints and maximizes the profit. Another example is in diet planning, where you want to find a diet that meets certain nutritional requirements while staying within a certain budget. This can be formulated as a system of inequalities, where the variables represent the amount of different foods to eat, and the constraints represent the nutritional requirements and budget constraints.
