To find out which point is a solution to the inequality [tex]\( y \leq 3x - 4 \)[/tex], we will evaluate the inequality at each of the given points.
Let's check each point step by step:
1. For the point [tex]\((-2,0)\)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality [tex]\( y \leq 3x - 4 \)[/tex]:
[tex]\[
0 \leq 3(-2) - 4
\][/tex]
[tex]\[
0 \leq -6 - 4
\][/tex]
[tex]\[
0 \leq -10
\][/tex]
This is not true, so [tex]\((-2,0)\)[/tex] is not a solution.
2. For the point [tex]\((3,1)\)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 1 \)[/tex] into the inequality [tex]\( y \leq 3x - 4 \)[/tex]:
[tex]\[
1 \leq 3(3) - 4
\][/tex]
[tex]\[
1 \leq 9 - 4
\][/tex]
[tex]\[
1 \leq 5
\][/tex]
This is true, so [tex]\((3,1)\)[/tex] is a solution.
3. For the point [tex]\((0,0)\)[/tex]:
- Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality [tex]\( y \leq 3x - 4 \)[/tex]:
[tex]\[
0 \leq 3(0) - 4
\][/tex]
[tex]\[
0 \leq 0 - 4
\][/tex]
[tex]\[
0 \leq -4
\][/tex]
This is not true, so [tex]\((0,0)\)[/tex] is not a solution.
4. For the point [tex]\((0,4)\)[/tex]:
- Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 4 \)[/tex] into the inequality [tex]\( y \leq 3x - 4 \)[/tex]:
[tex]\[
4 \leq 3(0) - 4
\][/tex]
[tex]\[
4 \leq 0 - 4
\][/tex]
[tex]\[
4 \leq -4
\][/tex]
This is not true, so [tex]\((0,4)\)[/tex] is not a solution.
Therefore, among the provided options, the only point that satisfies the inequality [tex]\( y \leq 3x - 4 \)[/tex] is [tex]\((3,1)\)[/tex]. Thus, the correct answer is:
B. [tex]\((3,1)\)[/tex]
