Choose the point which is a solution to the following system of inequalities:
[tex]\begin{array}{l}
x+2 y\ \textgreater \ 6 \
2 x+y\ \textgreater \ 3
\end{array}[/tex]
A. (0,4)
B. (-4,0)
C. (4,-4)
Answer (2)
Substitute each point into the inequalities.
Check if both inequalities are satisfied for each point.
Point ( 0 , 4 ) : 6"> 0 + 2 ( 4 ) > 6 and 3"> 2 ( 0 ) + 4 > 3 , so 6"> 8 > 6 and 3"> 4 > 3 . This point is a solution.
Point ( − 4 , 0 ) : 6"> − 4 + 2 ( 0 ) > 6 is false, so this point is not a solution.
Point ( 4 , − 4 ) : 6"> 4 + 2 ( − 4 ) > 6 is false, so this point is not a solution.
The solution is ( 0 , 4 ) .
Explanation
Understanding the Problem We are given the following system of inequalities:
6 \\ 2 x+y>3 \end{array}"> x + 2 y > 6 2 x + y > 3
We need to check which of the given points ( 0 , 4 ) , ( − 4 , 0 ) , and ( 4 , − 4 ) satisfies both inequalities. A point is a solution to the system if it satisfies both inequalities simultaneously.
Checking the point (0,4) Let's check the point ( 0 , 4 ) .
Substitute x = 0 and y = 4 into the first inequality:
6 \Rightarrow 8 > 6"> 0 + 2 ( 4 ) > 6 ⇒ 8 > 6
This inequality is satisfied.
Substitute x = 0 and y = 4 into the second inequality:
3 \Rightarrow 4 > 3"> 2 ( 0 ) + 4 > 3 ⇒ 4 > 3
This inequality is also satisfied. Therefore, ( 0 , 4 ) is a solution to the system.
Checking the point (-4,0) Let's check the point ( − 4 , 0 ) .
Substitute x = − 4 and y = 0 into the first inequality:
6 \Rightarrow -4 > 6"> − 4 + 2 ( 0 ) > 6 ⇒ − 4 > 6
This inequality is not satisfied. Therefore, ( − 4 , 0 ) is not a solution to the system.
Checking the point (4,-4) Let's check the point ( 4 , − 4 ) .
Substitute x = 4 and y = − 4 into the first inequality:
6 \Rightarrow 4 - 8 > 6 \Rightarrow -4 > 6"> 4 + 2 ( − 4 ) > 6 ⇒ 4 − 8 > 6 ⇒ − 4 > 6
This inequality is not satisfied. Therefore, ( 4 , − 4 ) is not a solution to the system.
Final Answer Only the point ( 0 , 4 ) satisfies both inequalities. Therefore, the solution to the system of inequalities is ( 0 , 4 ) .
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 different products, given constraints on resources like labor and materials. By graphing the inequalities, the company can find the feasible region, representing all possible production combinations that satisfy the constraints. The company can then identify the production levels that maximize profit within this feasible region.
The point (0, 4) is the only solution to the given system of inequalities, as it satisfies both inequalities. The points (-4, 0) and (4, -4) do not satisfy the inequalities. Therefore, the selected option is (0, 4).
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