Which of the points listed below are solutions for the system?
[tex]\begin{array}{l}
7 y\ \textgreater \ 7 x \\
3(x-2)\ \textless \ 0 \\
\text { I. }(-1,3) \\
\text { II. }(-5,9) \\
\text { III. }(0,0)
\end{array}[/tex]
a. I, II and III
b. I and II
c. I only
d. I and III
Answer (1)
Simplify the inequalities to x"> y > x and x < 2 .
Check point I. ( − 1 , 3 ) : -1"> 3 > − 1 and − 1 < 2 , so it satisfies both.
Check point II. ( − 5 , 9 ) : -5"> 9 > − 5 and − 5 < 2 , so it satisfies both.
Check point III. ( 0 , 0 ) : 0"> 0 > 0 is false, so it does not satisfy both.
The solutions are points I and II, therefore the answer is b .
Explanation
Understanding the Problem We are given a system of inequalities: 7x"> 7 y > 7 x 3 ( x − 2 ) < 0 and three points: I. ( − 1 , 3 ) II. ( − 5 , 9 ) III. ( 0 , 0 ) We need to determine which of these points satisfy both inequalities.
Simplifying the Inequalities First, let's simplify the inequalities. The first inequality, 7x"> 7 y > 7 x , can be simplified by dividing both sides by 7, resulting in: x"> y > x The second inequality, 3 ( x − 2 ) < 0 , can be simplified by dividing both sides by 3, resulting in: x − 2 < 0 Adding 2 to both sides, we get: x < 2
Checking the Points Now, we need to check which of the given points satisfy both inequalities: x"> y > x x < 2 For point I. ( − 1 , 3 ) :
Check if -1"> 3 > − 1 and − 1 < 2 .
-1"> 3 > − 1 is true, and − 1 < 2 is also true. So, point I satisfies both inequalities. For point II. ( − 5 , 9 ) :
Check if -5"> 9 > − 5 and − 5 < 2 .
-5"> 9 > − 5 is true, and − 5 < 2 is also true. So, point II satisfies both inequalities. For point III. ( 0 , 0 ) :
Check if 0"> 0 > 0 and 0 < 2 .
0"> 0 > 0 is false, and 0 < 2 is true. Since the first inequality is not satisfied, point III does not satisfy both inequalities.
Conclusion Based on the above analysis: Point I. ( − 1 , 3 ) satisfies both inequalities. Point II. ( − 5 , 9 ) satisfies both inequalities. Point III. ( 0 , 0 ) does not satisfy both inequalities. Therefore, the points that are solutions to the system are I and II.
Examples
Systems of inequalities are used in various real-world applications, such as in economics to determine feasible production regions given resource constraints, in engineering to design structures within safety limits, and in everyday life to manage budgets and time. For example, suppose you have a budget constraint and a time constraint for a project. The system of inequalities can help you determine if a certain allocation of money and time is feasible for completing the project.
