To determine which of the given inequalities the point [tex]\((0,0)\)[/tex] satisfies, we will substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into each inequality and check if it results in a true statement.
Checking Inequality A:
[tex]\[ y + 7 < 2x - 6 \][/tex]
Substitute [tex]\((x, y) = (0, 0)\)[/tex]:
[tex]\[ 0 + 7 < 2(0) - 6 \][/tex]
[tex]\[ 7 < -6 \][/tex]
This statement is false. Therefore, the point [tex]\((0,0)\)[/tex] does not satisfy [tex]\(y + 7 < 2x - 6\)[/tex].
Checking Inequality B:
[tex]\[ y - 7 < 2x - 6 \][/tex]
Substitute [tex]\((x, y) = (0, 0)\)[/tex]:
[tex]\[ 0 - 7 < 2(0) - 6 \][/tex]
[tex]\[ -7 < -6 \][/tex]
This statement is true. Therefore, the point [tex]\((0,0)\)[/tex] satisfies [tex]\(y - 7 < 2x - 6\)[/tex].
Checking Inequality C:
[tex]\[ y + 7 < 2x + 6 \][/tex]
Substitute [tex]\((x, y) = (0, 0)\)[/tex]:
[tex]\[ 0 + 7 < 2(0) + 6 \][/tex]
[tex]\[ 7 < 6 \][/tex]
This statement is false. Therefore, the point [tex]\((0,0)\)[/tex] does not satisfy [tex]\(y + 7 < 2x + 6\)[/tex].
Checking Inequality D:
[tex]\[ y - 6 < 2x - 7 \][/tex]
Substitute [tex]\((x, y) = (0, 0)\)[/tex]:
[tex]\[ 0 - 6 < 2(0) - 7 \][/tex]
[tex]\[ -6 < -7 \][/tex]
This statement is false. Therefore, the point [tex]\((0,0)\)[/tex] does not satisfy [tex]\(y - 6 < 2x - 7\)[/tex].
Summary:
The point [tex]\((0,0)\)[/tex] satisfies only inequality B:
[tex]\[ y - 7 < 2x - 6 \][/tex]
Therefore, the correct answer is:
B. [tex]\( y - 7 < 2 x - 6 \)[/tex]
