Which of the below points is in the solution region to the following system of inequalities?
[tex]
\begin{array}{l}
y \geq-x+5 \\
-x \leq-2 y-3
\end{array}
[/tex]
(-3,-3)
(0,0)
(10,0)
(4,6)
Answer (2)
Check each point against the inequalities.
Point ( − 3 , − 3 ) : − 3 g e − ( − 3 ) + 5 is false.
Point ( 0 , 0 ) : 0 g e − 0 + 5 is false.
Point ( 10 , 0 ) : 0 g e − 10 + 5 is true, but − 10 g e − 2 ( 0 ) − 3 is false.
Point ( 4 , 6 ) : 6 g e − 4 + 5 and − 4 g e − 2 ( 6 ) − 3 are both true.
The point in the solution region is ( 4 , 6 ) .
Explanation
Analyze the problem We are given the following system of inequalities:
y g e − x + 5
− xg e − 2 y − 3
We need to check which of the given points satisfies both inequalities.
Check point (-3, -3) Let's check the point ( − 3 , − 3 ) .
For the first inequality, we have:
− 3 g e − ( − 3 ) + 5
− 3 g e 3 + 5
− 3 g e 8 , which is false.
Since the first inequality is not satisfied, we don't need to check the second inequality.
Check point (0, 0) Let's check the point ( 0 , 0 ) .
For the first inequality, we have:
0 g e − 0 + 5
0 g e 5 , which is false.
Since the first inequality is not satisfied, we don't need to check the second inequality.
Check point (10, 0) Let's check the point ( 10 , 0 ) .
For the first inequality, we have:
0 g e − 10 + 5
0 g e − 5 , which is true.
For the second inequality, we have:
− 10 g e − 2 ( 0 ) − 3
− 10 g e − 3 , which is false.
Since the second inequality is not satisfied, the point ( 10 , 0 ) is not in the solution region.
Check point (4, 6) Let's check the point ( 4 , 6 ) .
For the first inequality, we have:
6 g e − 4 + 5
6 g e 1 , which is true.
For the second inequality, we have:
− 4 g e − 2 ( 6 ) − 3
− 4 g e − 12 − 3
− 4 g e − 15 , which is true.
Since both inequalities are satisfied, the point ( 4 , 6 ) is in the solution region.
Final Answer Therefore, the point ( 4 , 6 ) is in the solution region to the given system of inequalities.
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 resources like labor, materials, and production capacity. The solution region represents the set of feasible solutions that satisfy all the constraints.
To find the point in the solution region of the given inequalities, each point was checked. The point ( 4 , 6 ) satisfied both inequalities, making it the correct answer. Therefore, the point in the solution region is ( 4 , 6 ) .
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