The point $(4,1)$ is a solution to the following system of inequalities:
[tex]
\begin{array}{l}
2 x+y\ \textgreater \ 3 \
3 x-y\ \textless \ 1
\end{array}
[/tex]
True or False?
Answer (2)
Substitute the point ( 4 , 1 ) into the first inequality 3"> 2 x + y > 3 and check if it is true: 3 \Rightarrow 9 > 3"> 2 ( 4 ) + 1 > 3 ⇒ 9 > 3 , which is true.
Substitute the point ( 4 , 1 ) into the second inequality 3 x − y < 1 and check if it is true: 3 ( 4 ) − 1 < 1 ⇒ 11 < 1 , which is false.
Since the point ( 4 , 1 ) does not satisfy both inequalities, it is not a solution to the system of inequalities.
Therefore, the statement is False .
Explanation
Problem Setup and Goal We are given the point ( 4 , 1 ) and the system of inequalities:
3 \\ 3 x-y<1 \end{array}"> 2 x + y > 3 3 x − y < 1
We need to determine if the point ( 4 , 1 ) is a solution to the system of inequalities. To do this, we will substitute x = 4 and y = 1 into each inequality and check if the inequalities hold true.
Checking the First Inequality First, let's substitute x = 4 and y = 1 into the first inequality:
3"> 2 x + y > 3 3"> 2 ( 4 ) + 1 > 3 3"> 8 + 1 > 3 3"> 9 > 3
Since 3"> 9 > 3 is true, the point ( 4 , 1 ) satisfies the first inequality.
Checking the Second Inequality Now, let's substitute x = 4 and y = 1 into the second inequality:
3 x − y < 1 3 ( 4 ) − 1 < 1 12 − 1 < 1 11 < 1
Since 11 < 1 is false, the point ( 4 , 1 ) does not satisfy the second inequality.
Conclusion For the point ( 4 , 1 ) to be a solution to the system of inequalities, it must satisfy both inequalities. Since the point ( 4 , 1 ) satisfies the first inequality but does not satisfy the second inequality, it is not a solution to the system of inequalities.
Therefore, the statement "The point ( 4 , 1 ) is a solution to the following system of inequalities: 3 \\ 3 x-y<1 \end{array}"> 2 x + y > 3 3 x − y < 1 " is False.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, which helps optimize solutions in business and economics. 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 company can then identify the production levels that maximize profit within this feasible region. This approach ensures efficient resource allocation and optimal decision-making.
The point ( 4 , 1 ) satisfies the first inequality 3"> 2 x + y > 3 , but does not satisfy the second inequality 3 x − y < 1 . Therefore, it is not a solution to the system of inequalities, making the statement False. So, the answer is False .
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