To determine which points lie in the solution set of the given system of inequalities, we need to check each point individually against all the inequalities:
[tex]\[
\begin{array}{l}
x-y \geq -4 \\
2x-y \leq 5 \\
2y+x > 1
\end{array}
\][/tex]
Point [tex]\((-2, 3)\)[/tex]:
1. Check [tex]\( x - y \geq -4 \)[/tex]:
[tex]\[
-2 - 3 \geq -4 \quad \Rightarrow \quad -5 \geq -4 \quad \text{False}
\][/tex]
Since this point does not satisfy the first inequality, it is not in the solution set.
Point [tex]\((-1, 3)\)[/tex]:
1. Check [tex]\( x - y \geq -4 \)[/tex]:
[tex]\[
-1 - 3 \geq -4 \quad \Rightarrow \quad -4 \geq -4 \quad \text{True}
\][/tex]
2. Check [tex]\( 2x - y \leq 5 \)[/tex]:
[tex]\[
2(-1) - 3 \leq 5 \quad \Rightarrow \quad -2 - 3 \leq 5 \quad \Rightarrow \quad -5 \leq 5 \quad \text{True}
\][/tex]
3. Check [tex]\( 2y + x > 1 \)[/tex]:
[tex]\[
2(3) + (-1) > 1 \quad \Rightarrow \quad 6 - 1 > 1 \quad \Rightarrow \quad 5 > 1 \quad \text{True}
\][/tex]
Since this point satisfies all the inequalities, it is in the solution set.
Point [tex]\((0, 0)\)[/tex]:
1. Check [tex]\( x - y \geq -4 \)[/tex]:
[tex]\[
0 - 0 \geq -4 \quad \Rightarrow \quad 0 \geq -4 \quad \text{True}
\][/tex]
2. Check [tex]\( 2x - y \leq 5 \)[/tex]:
[tex]\[
2(0) - 0 \leq 5 \quad \Rightarrow \quad 0 \leq 5 \quad \text{True}
\][/tex]
3. Check [tex]\( 2y + x > 1 \)[/tex]:
[tex]\[
2(0) + 0 > 1 \quad \Rightarrow \quad 0 > 1 \quad \text{False}
\][/tex]
Since this point does not satisfy the third inequality, it is not in the solution set.
Point [tex]\((2, -2)\)[/tex]:
1. Check [tex]\( x - y \geq -4 \)[/tex]:
[tex]\[
2 - (-2) \geq -4 \quad \Rightarrow \quad 2 + 2 \geq -4 \quad \Rightarrow \quad 4 \geq -4 \quad \text{True}
\][/tex]
2. Check [tex]\( 2x - y \leq 5 \)[/tex]:
[tex]\[
2(2) - (-2) \leq 5 \quad \Rightarrow \quad 4 + 2 \leq 5 \quad \Rightarrow \quad 6 \leq 5 \quad \text{False}
\][/tex]
Since this point does not satisfy the second inequality, it is not in the solution set.
Point [tex]\((3, 5)\)[/tex]:
1. Check [tex]\( x - y \geq -4 \)[/tex]:
[tex]\[
3 - 5 \geq -4 \quad \Rightarrow \quad -2 \geq -4 \quad \text{True}
\][/tex]
2. Check [tex]\( 2x - y \leq 5 \)[/tex]:
[tex]\[
2(3) - 5 \leq 5 \quad \Rightarrow \quad 6 - 5 \leq 5 \quad \Rightarrow \quad 1 \leq 5 \quad \text{True}
\][/tex]
3. Check [tex]\( 2y + x > 1 \)[/tex]:
[tex]\[
2(5) + 3 > 1 \quad \Rightarrow \quad 10 + 3 > 1 \quad \Rightarrow \quad 13 > 1 \quad \text{True}
\][/tex]
Since this point satisfies all the inequalities, it is in the solution set.
Summary:
The points that lie in the solution set of the given system of inequalities are:
- [tex]\((-1, 3)\)[/tex]
- [tex]\((3, 5)\)[/tex]
