Which of the below points is in the solution region to the following system of inequalities?
[tex]$\begin{array}{l}
2 x-y\ \textless \ 1 \
x+2 y\ \textless \ -4
\end{array}$[/tex]
(-8,-1)
(-2,4)
(0,6)
(4,0)
Answer (2)
Substitute each point into the inequalities.
Check if both inequalities are true for each point.
Point ( − 8 , − 1 ) : 2 ( − 8 ) − ( − 1 ) < 1 and ( − 8 ) + 2 ( − 1 ) < − 4 are both true.
The solution is ( − 8 , − 1 ) .
Explanation
Analyze the problem We are given a system of two inequalities:
2 x − y < 1 x + 2 y < − 4
We need to determine which of the given points satisfies both inequalities. To do this, we will substitute the x and y values of each point into both inequalities and check if the inequalities hold true.
Test the point (-8, -1) Let's test the point ( − 8 , − 1 ) :
For the first inequality: 2 ( − 8 ) − ( − 1 ) < 1 − 16 + 1 < 1 − 15 < 1 This is true.
For the second inequality: ( − 8 ) + 2 ( − 1 ) < − 4 − 8 − 2 < − 4 − 10 < − 4 This is true as well. Since both inequalities are true for the point ( − 8 , − 1 ) , this point is in the solution region.
Test the point (-2, 4) Let's test the point ( − 2 , 4 ) :
For the first inequality: 2 ( − 2 ) − ( 4 ) < 1 − 4 − 4 < 1 − 8 < 1 This is true.
For the second inequality: ( − 2 ) + 2 ( 4 ) < − 4 − 2 + 8 < − 4 6 < − 4 This is false. Since the second inequality is false for the point ( − 2 , 4 ) , this point is not in the solution region.
Test the point (0, 6) Let's test the point ( 0 , 6 ) :
For the first inequality: 2 ( 0 ) − ( 6 ) < 1 0 − 6 < 1 − 6 < 1 This is true.
For the second inequality: ( 0 ) + 2 ( 6 ) < − 4 0 + 12 < − 4 12 < − 4 This is false. Since the second inequality is false for the point ( 0 , 6 ) , this point is not in the solution region.
Test the point (4, 0) Let's test the point ( 4 , 0 ) :
For the first inequality: 2 ( 4 ) − ( 0 ) < 1 8 − 0 < 1 8 < 1 This is false.
For the second inequality: ( 4 ) + 2 ( 0 ) < − 4 4 + 0 < − 4 4 < − 4 This is false. Since both inequalities are false for the point ( 4 , 0 ) , this point is not in the solution region.
Final Answer Only the point ( − 8 , − 1 ) satisfies both inequalities. Therefore, the point ( − 8 , − 1 ) is in the solution region.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where they help optimize solutions under constraints. For example, a company might use a system of inequalities to determine the optimal production levels of different products, given constraints on resources like labor and materials. Similarly, diet planning can use inequalities to ensure nutritional requirements are met within certain calorie limits. These applications demonstrate how systems of inequalities provide a structured approach to problem-solving in diverse fields.
None of the tested points (-8,-1), (-2,4), (0,6), or (4,0) satisfy both inequalities. Therefore, none of these points are in the solution region of the system. The answer is that none of the provided points are valid solutions.
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