$\begin{array}{l}
y \leq-x+1 \\
y>x
\end{array}$

(-3,5)
(-2,2)
(-1,-3)
(0,-1)

Substitute each point into the inequalities.
Check if both inequalities are true for each point.
The point ( − 2 , 2 ) satisfies both inequalities: 2 ≤ − ( − 2 ) + 1 and -2"> 2 > − 2 .
Therefore, the solution is ( − 2 , 2 ) ​ .

Explanation

Problem Analysis We are given a system of inequalities and a set of points. Our goal is to determine which points satisfy both inequalities. The inequalities are:

y ≤ − x + 1 x"> y > x
The points are: ( − 3 , 5 ) , ( − 2 , 2 ) , ( − 1 , − 3 ) , ( 0 , − 1 ) . We will substitute each point into both inequalities to see if they hold true.

Testing (-3,5) Let's test the point ( − 3 , 5 ) :

For the first inequality, y ≤ − x + 1 , we have: 5 ≤ − ( − 3 ) + 1 5 ≤ 3 + 1 5 ≤ 4 This is false.
For the second inequality, x"> y > x , we have: -3"> 5 > − 3 This is true.
Since the first inequality is false, the point ( − 3 , 5 ) does not satisfy the system of inequalities.

Testing (-2,2) Let's test the point ( − 2 , 2 ) :

For the first inequality, y ≤ − x + 1 , we have: 2 ≤ − ( − 2 ) + 1 2 ≤ 2 + 1 2 ≤ 3 This is true.
For the second inequality, x"> y > x , we have: -2"> 2 > − 2 This is true.
Since both inequalities are true, the point ( − 2 , 2 ) satisfies the system of inequalities.

Testing (-1,-3) Let's test the point ( − 1 , − 3 ) :

For the first inequality, y ≤ − x + 1 , we have: − 3 ≤ − ( − 1 ) + 1 − 3 ≤ 1 + 1 − 3 ≤ 2 This is true.
For the second inequality, x"> y > x , we have: -1"> − 3 > − 1 This is false.
Since the second inequality is false, the point ( − 1 , − 3 ) does not satisfy the system of inequalities.

Testing (0,-1) Let's test the point ( 0 , − 1 ) :

For the first inequality, y ≤ − x + 1 , we have: − 1 ≤ − ( 0 ) + 1 − 1 ≤ 0 + 1 − 1 ≤ 1 This is true.
For the second inequality, x"> y > x , we have: 0"> − 1 > 0 This is false.
Since the second inequality is false, the point ( 0 , − 1 ) does not satisfy the system of inequalities.

Conclusion Therefore, only the point ( − 2 , 2 ) satisfies both 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 and materials. Each constraint can be represented as an inequality, and the solution to the system of inequalities represents the feasible region where all constraints are satisfied. In this case, we found which points satisfy the given constraints, which is a fundamental step in solving optimization problems.

Answered by GinnyAnswer | 2025-07-07