To determine which points satisfy the given system of inequalities, we need to check each point against the three inequalities:
[tex]\[
\begin{array}{l}
y < x \\
y < 5 \\
x < 4
\end{array}
\][/tex]
Let's check each given point one-by-one:
Option A: [tex]\((3, 0)\)[/tex]
1. [tex]\(y < x\)[/tex]: [tex]\(0 < 3\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(0 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(3 < 4\)[/tex] (True)
Since all three inequalities are satisfied, [tex]\((3, 0)\)[/tex] is a solution.
Option B: [tex]\((3, 4)\)[/tex]
1. [tex]\(y < x\)[/tex]: [tex]\(4 < 3\)[/tex] (False)
2. [tex]\(y < 5\)[/tex]: [tex]\(4 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(3 < 4\)[/tex] (True)
Since the first inequality is not satisfied, [tex]\((3, 4)\)[/tex] is not a solution.
Option C: [tex]\((6, 2)\)[/tex]
1. [tex]\(y < x\)[/tex]: [tex]\(2 < 6\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(2 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(6 < 4\)[/tex] (False)
Since the third inequality is not satisfied, [tex]\((6, 2)\)[/tex] is not a solution.
Option D: [tex]\((2, 1)\)[/tex]
1. [tex]\(y < x\)[/tex]: [tex]\(1 < 2\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(1 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(2 < 4\)[/tex] (True)
Since all three inequalities are satisfied, [tex]\((2, 1)\)[/tex] is a solution.
Option E: [tex]\((0, -1)\)[/tex]
1. [tex]\(y < x\)[/tex]: [tex]\(-1 < 0\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(-1 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(0 < 4\)[/tex] (True)
Since all three inequalities are satisfied, [tex]\((0, -1)\)[/tex] is a solution.
Option F: [tex]\((1, 7)\)[/tex]
1. [tex]\(y < x\)[/tex]: [tex]\(7 < 1\)[/tex] (False)
2. [tex]\(y < 5\)[/tex]: [tex]\(7 < 5\)[/tex] (False)
3. [tex]\(x < 4\)[/tex]: [tex]\(1 < 4\)[/tex] (True)
Since the first and second inequalities are not satisfied, [tex]\((1, 7)\)[/tex] is not a solution.
After checking all points, the solutions to the system of inequalities are:
A. [tex]\((3, 0)\)[/tex], D. [tex]\((2, 1)\)[/tex], E. [tex]\((0, -1)\)[/tex]
